# 梯度向量基礎 · 脈衝 R06

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261015-advance-20261113-r06-gradient-vector-foundations · 2026-10-15 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

[欄序與來源綁定資料](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[1,[2,1]],[2,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0001/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 2. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[2,[2,1,0]],[3,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0002/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 3. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[3,[2,1]],[4,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0003/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 4. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[4,[2,1,0]],[5,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0004/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 5. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[5,[2,1]],[6,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0005/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 6. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[6,[2,1,0]],[7,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0006/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 7. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[7,[2,1]],[8,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0007/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 8. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[8,[2,1,0]],[9,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0008/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 9. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[9,[2,1]],[10,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0009/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 10. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[10,[2,1,0]],[11,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0010/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 11. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[11,[2,1]],[12,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0011/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 12. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[12,[2,1,0]],[13,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0012/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 13. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[13,[2,1]],[14,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0013/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 14. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[14,[2,1,0]],[15,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0014/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 15. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[15,[2,1]],[16,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0015/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 16. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[16,[2,1,0]],[17,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0016/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 17. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[17,[2,1]],[18,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0017/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 18. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[18,[2,1,0]],[19,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0018/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 19. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[19,[2,1]],[20,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0019/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 20. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[20,[2,1,0]],[21,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0020/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 21. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[21,[2,1]],[22,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0021/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 22. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[22,[2,1,0]],[23,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0022/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 23. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[23,[2,1]],[24,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0023/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 24. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":3,"terms":[[24,[2,1,0]],[25,[0,1,1]],[1,[0,0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0024/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 25. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-symbolic-polynomial；資料={"dimension":2,"terms":[[25,[2,1]],[26,[0,1]],[1,[0,0]]]}。請依 [dimension, canonical_terms, coordinate_derivative_polynomials, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)

Wave6第七層核對=(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], standard_Euclidean_coordinate_order_preserved)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0025/)

梯度向量 · qf-mod44-gradient-symbolic-polynomial · candidate





## 26. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[1,[2,1]],[2,[0,1]],[1,[0,0]]],"point":[[1,1],[2,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [1, 1], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], [2, 3], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [2, [0, 1]], [1, [2, 1]]], [1, 1], [[[2, [1, 1]]], [[2, [0, 0]], [1, [2, 0]]]], [2, 3], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0026/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 27. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[2,[2,1,0]],[3,[0,1,1]],[1,[0,0,0]]],"point":[[2,1],[3,2],[4,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [3, [0, 1, 1]], [2, [2, 1, 0]]], [2, 3/2, 4/3], [[[4, [1, 1, 0]]], [[3, [0, 0, 1]], [2, [2, 0, 0]]], [[3, [0, 1, 0]]]], [12, 12, 9/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0027/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 28. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[3,[2,1]],[4,[0,1]],[1,[0,0]]],"point":[[3,1],[4,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [4, [0, 1]], [3, [2, 1]]], [3, 2], [[[6, [1, 1]]], [[4, [0, 0]], [3, [2, 0]]]], [36, 31], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0028/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 29. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[4,[2,1,0]],[5,[0,1,1]],[1,[0,0,0]]],"point":[[4,1],[5,2],[6,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [4, 5/2, 2], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], [80, 74, 25/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [5, [0, 1, 1]], [4, [2, 1, 0]]], [4, 5/2, 2], [[[8, [1, 1, 0]]], [[5, [0, 0, 1]], [4, [2, 0, 0]]], [[5, [0, 1, 0]]]], [80, 74, 25/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0029/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 30. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[5,[2,1]],[6,[0,1]],[1,[0,0]]],"point":[[5,1],[6,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [5, 3], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], [150, 131], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [6, [0, 1]], [5, [2, 1]]], [5, 3], [[[10, [1, 1]]], [[6, [0, 0]], [5, [2, 0]]]], [150, 131], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0030/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 31. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[6,[2,1,0]],[7,[0,1,1]],[1,[0,0,0]]],"point":[[6,1],[7,2],[8,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [6, 7/2, 8/3], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], [252, 704/3, 49/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [7, [0, 1, 1]], [6, [2, 1, 0]]], [6, 7/2, 8/3], [[[12, [1, 1, 0]]], [[7, [0, 0, 1]], [6, [2, 0, 0]]], [[7, [0, 1, 0]]]], [252, 704/3, 49/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0031/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 32. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[7,[2,1]],[8,[0,1]],[1,[0,0]]],"point":[[7,1],[8,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [7, 4], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], [392, 351], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [8, [0, 1]], [7, [2, 1]]], [7, 4], [[[14, [1, 1]]], [[8, [0, 0]], [7, [2, 0]]]], [392, 351], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0032/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 33. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[8,[2,1,0]],[9,[0,1,1]],[1,[0,0,0]]],"point":[[8,1],[9,2],[10,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [8, 9/2, 10/3], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], [576, 542, 81/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [9, [0, 1, 1]], [8, [2, 1, 0]]], [8, 9/2, 10/3], [[[16, [1, 1, 0]]], [[9, [0, 0, 1]], [8, [2, 0, 0]]], [[9, [0, 1, 0]]]], [576, 542, 81/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0033/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 34. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[9,[2,1]],[10,[0,1]],[1,[0,0]]],"point":[[9,1],[10,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [9, 5], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], [810, 739], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [10, [0, 1]], [9, [2, 1]]], [9, 5], [[[18, [1, 1]]], [[10, [0, 0]], [9, [2, 0]]]], [810, 739], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0034/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 35. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[10,[2,1,0]],[11,[0,1,1]],[1,[0,0,0]]],"point":[[10,1],[11,2],[12,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [10, 11/2, 4], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], [1100, 1044, 121/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [11, [0, 1, 1]], [10, [2, 1, 0]]], [10, 11/2, 4], [[[20, [1, 1, 0]]], [[11, [0, 0, 1]], [10, [2, 0, 0]]], [[11, [0, 1, 0]]]], [1100, 1044, 121/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0035/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 36. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[11,[2,1]],[12,[0,1]],[1,[0,0]]],"point":[[11,1],[12,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [11, 6], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], [1452, 1343], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [12, [0, 1]], [11, [2, 1]]], [11, 6], [[[22, [1, 1]]], [[12, [0, 0]], [11, [2, 0]]]], [1452, 1343], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0036/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 37. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[12,[2,1,0]],[13,[0,1,1]],[1,[0,0,0]]],"point":[[12,1],[13,2],[14,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [12, 13/2, 14/3], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], [1872, 5366/3, 169/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [13, [0, 1, 1]], [12, [2, 1, 0]]], [12, 13/2, 14/3], [[[24, [1, 1, 0]]], [[13, [0, 0, 1]], [12, [2, 0, 0]]], [[13, [0, 1, 0]]]], [1872, 5366/3, 169/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0037/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 38. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[13,[2,1]],[14,[0,1]],[1,[0,0]]],"point":[[13,1],[14,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [13, 7], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], [2366, 2211], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [14, [0, 1]], [13, [2, 1]]], [13, 7], [[[26, [1, 1]]], [[14, [0, 0]], [13, [2, 0]]]], [2366, 2211], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0038/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 39. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[14,[2,1,0]],[15,[0,1,1]],[1,[0,0,0]]],"point":[[14,1],[15,2],[16,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [14, 15/2, 16/3], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], [2940, 2824, 225/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [15, [0, 1, 1]], [14, [2, 1, 0]]], [14, 15/2, 16/3], [[[28, [1, 1, 0]]], [[15, [0, 0, 1]], [14, [2, 0, 0]]], [[15, [0, 1, 0]]]], [2940, 2824, 225/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0039/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 40. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[15,[2,1]],[16,[0,1]],[1,[0,0]]],"point":[[15,1],[16,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [16, [0, 1]], [15, [2, 1]]], [15, 8], [[[30, [1, 1]]], [[16, [0, 0]], [15, [2, 0]]]], [3600, 3391], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0040/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 41. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[16,[2,1,0]],[17,[0,1,1]],[1,[0,0,0]]],"point":[[16,1],[17,2],[18,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [16, 17/2, 6], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], [4352, 4198, 289/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [17, [0, 1, 1]], [16, [2, 1, 0]]], [16, 17/2, 6], [[[32, [1, 1, 0]]], [[17, [0, 0, 1]], [16, [2, 0, 0]]], [[17, [0, 1, 0]]]], [4352, 4198, 289/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0041/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 42. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[17,[2,1]],[18,[0,1]],[1,[0,0]]],"point":[[17,1],[18,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [17, 9], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], [5202, 4931], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [18, [0, 1]], [17, [2, 1]]], [17, 9], [[[34, [1, 1]]], [[18, [0, 0]], [17, [2, 0]]]], [5202, 4931], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0042/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 43. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[18,[2,1,0]],[19,[0,1,1]],[1,[0,0,0]]],"point":[[18,1],[19,2],[20,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [19, [0, 1, 1]], [18, [2, 1, 0]]], [18, 19/2, 20/3], [[[36, [1, 1, 0]]], [[19, [0, 0, 1]], [18, [2, 0, 0]]], [[19, [0, 1, 0]]]], [6156, 17876/3, 361/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0043/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 44. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[19,[2,1]],[20,[0,1]],[1,[0,0]]],"point":[[19,1],[20,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [19, 10], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], [7220, 6879], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [20, [0, 1]], [19, [2, 1]]], [19, 10], [[[38, [1, 1]]], [[20, [0, 0]], [19, [2, 0]]]], [7220, 6879], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0044/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 45. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[20,[2,1,0]],[21,[0,1,1]],[1,[0,0,0]]],"point":[[20,1],[21,2],[22,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [20, 21/2, 22/3], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], [8400, 8154, 441/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [21, [0, 1, 1]], [20, [2, 1, 0]]], [20, 21/2, 22/3], [[[40, [1, 1, 0]]], [[21, [0, 0, 1]], [20, [2, 0, 0]]], [[21, [0, 1, 0]]]], [8400, 8154, 441/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0045/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 46. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[21,[2,1]],[22,[0,1]],[1,[0,0]]],"point":[[21,1],[22,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [21, 11], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], [9702, 9283], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [22, [0, 1]], [21, [2, 1]]], [21, 11], [[[42, [1, 1]]], [[22, [0, 0]], [21, [2, 0]]]], [9702, 9283], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0046/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 47. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[22,[2,1,0]],[23,[0,1,1]],[1,[0,0,0]]],"point":[[22,1],[23,2],[24,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [22, 23/2, 8], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], [11132, 10832, 529/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [23, [0, 1, 1]], [22, [2, 1, 0]]], [22, 23/2, 8], [[[44, [1, 1, 0]]], [[23, [0, 0, 1]], [22, [2, 0, 0]]], [[23, [0, 1, 0]]]], [11132, 10832, 529/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0047/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 48. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[23,[2,1]],[24,[0,1]],[1,[0,0]]],"point":[[23,1],[24,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [23, 12], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], [12696, 12191], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [24, [0, 1]], [23, [2, 1]]], [23, 12], [[[46, [1, 1]]], [[24, [0, 0]], [23, [2, 0]]]], [12696, 12191], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0048/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 49. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":3,"terms":[[24,[2,1,0]],[25,[0,1,1]],[1,[0,0,0]]],"point":[[24,1],[25,2],[26,3]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [24, 25/2, 26/3], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], [14400, 42122/3, 625/2], exact_rational_coordinate_evaluation)

Wave6第七層核對=(3, [[1, [0, 0, 0]], [25, [0, 1, 1]], [24, [2, 1, 0]]], [24, 25/2, 26/3], [[[48, [1, 1, 0]]], [[25, [0, 0, 1]], [24, [2, 0, 0]]], [[25, [0, 1, 0]]]], [14400, 42122/3, 625/2], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0049/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 50. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-rational-point；資料={"dimension":2,"terms":[[25,[2,1]],[26,[0,1]],[1,[0,0]]],"point":[[25,1],[26,2]]}。請依 [dimension, canonical_terms, point, coordinate_derivative_polynomials, gradient_values, certificate] 欄序回答。

查看答案與解釋(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [25, 13], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], [16250, 15651], exact_rational_coordinate_evaluation)

Wave6第七層核對=(2, [[1, [0, 0]], [26, [0, 1]], [25, [2, 1]]], [25, 13], [[[50, [1, 1]]], [[26, [0, 0]], [25, [2, 0]]]], [16250, 15651], exact_rational_coordinate_evaluation)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0050/)

梯度向量 · qf-mod44-gradient-rational-point · candidate





## 51. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[1,[2,0]],[2,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0051/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 52. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[2,[2,0]],[3,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0052/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 53. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[3,[2,0]],[4,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0053/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 54. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[4,[2,0]],[5,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0054/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 55. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[5,[2,0]],[6,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0055/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 56. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[6,[2,0]],[7,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0056/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 57. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[7,[2,0]],[8,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0057/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 58. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[8,[2,0]],[9,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0058/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 59. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[9,[2,0]],[10,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0059/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 60. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[10,[2,0]],[11,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0060/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 61. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[11,[2,0]],[12,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0061/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 62. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[12,[2,0]],[13,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0062/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 63. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[13,[2,0]],[14,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0063/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 64. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[14,[2,0]],[15,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0064/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 65. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[15,[2,0]],[16,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0065/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 66. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[16,[2,0]],[17,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0066/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 67. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[17,[2,0]],[18,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0067/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 68. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[18,[2,0]],[19,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0068/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 69. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[19,[2,0]],[20,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0069/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 70. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[20,[2,0]],[21,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0070/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 71. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[21,[2,0]],[22,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0071/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 72. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[22,[2,0]],[23,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0072/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 73. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[23,[2,0]],[24,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0073/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 74. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[24,[2,0]],[25,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0074/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 75. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-stationary-audit；資料={"dimension":2,"terms":[[25,[2,0]],[26,[0,2]]],"point":[0,0]}。請依 [dimension, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

最後固定的否是不提供extremum證書，不否定某個特殊函數（例如常數函數）實際具有極值。

查看答案與解釋(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)

Wave6第七層核對=(2, [0, 0], [0, 0], 是, 否, gradient_zero_means_stationary_not_necessarily_extremum)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0075/)

梯度向量 · qf-mod44-gradient-stationary-audit · candidate





## 76. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"incomplete_coordinate_list","scale":1}。請依 [mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(incomplete_coordinate_list, 3, [1*partial_x, 1*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)

Wave6第七層核對=(incomplete_coordinate_list, 3, [1*partial_x, 1*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0076/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 77. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"nondifferentiable_norm","scale":2}。請依 [mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)

Wave6第七層核對=(nondifferentiable_norm, f(x,y)=2*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0077/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 78. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"partials_exist_not_differentiable","scale":3}。請依 [mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(partials_exist_not_differentiable, f(x,y)=3*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [3, 0], 是, along_y=x:f(x,x)=3/2*x_but_linear_candidate=3*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)

Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=3*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [3, 0], 是, along_y=x:f(x,x)=3/2*x_but_linear_candidate=3*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0078/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 79. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"saddle_stationary","scale":4}。請依 [mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(saddle_stationary, f(x,y)=4*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)

Wave6第七層核對=(saddle_stationary, f(x,y)=4*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0079/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 80. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"incomplete_coordinate_list","scale":5}。請依 [mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(incomplete_coordinate_list, 3, [5*partial_x, 5*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)

Wave6第七層核對=(incomplete_coordinate_list, 3, [5*partial_x, 5*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0080/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 81. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"nondifferentiable_norm","scale":6}。請依 [mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(nondifferentiable_norm, f(x,y)=6*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)

Wave6第七層核對=(nondifferentiable_norm, f(x,y)=6*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0081/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 82. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"partials_exist_not_differentiable","scale":7}。請依 [mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(partials_exist_not_differentiable, f(x,y)=7*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [7, 0], 是, along_y=x:f(x,x)=7/2*x_but_linear_candidate=7*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)

Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=7*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [7, 0], 是, along_y=x:f(x,x)=7/2*x_but_linear_candidate=7*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0082/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 83. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"saddle_stationary","scale":8}。請依 [mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)

Wave6第七層核對=(saddle_stationary, f(x,y)=8*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0083/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 84. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"incomplete_coordinate_list","scale":9}。請依 [mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)

Wave6第七層核對=(incomplete_coordinate_list, 3, [9*partial_x, 9*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0084/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 85. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"nondifferentiable_norm","scale":10}。請依 [mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)

Wave6第七層核對=(nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0085/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 86. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"partials_exist_not_differentiable","scale":11}。請依 [mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], 是, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)

Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=11*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [11, 0], 是, along_y=x:f(x,x)=11/2*x_but_linear_candidate=11*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0086/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 87. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"saddle_stationary","scale":12}。請依 [mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(saddle_stationary, f(x,y)=12*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)

Wave6第七層核對=(saddle_stationary, f(x,y)=12*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0087/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 88. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"incomplete_coordinate_list","scale":13}。請依 [mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(incomplete_coordinate_list, 3, [13*partial_x, 13*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)

Wave6第七層核對=(incomplete_coordinate_list, 3, [13*partial_x, 13*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0088/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 89. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"nondifferentiable_norm","scale":14}。請依 [mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(nondifferentiable_norm, f(x,y)=14*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)

Wave6第七層核對=(nondifferentiable_norm, f(x,y)=14*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0089/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 90. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"partials_exist_not_differentiable","scale":15}。請依 [mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(partials_exist_not_differentiable, f(x,y)=15*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [15, 0], 是, along_y=x:f(x,x)=15/2*x_but_linear_candidate=15*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)

Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=15*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [15, 0], 是, along_y=x:f(x,x)=15/2*x_but_linear_candidate=15*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0090/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 91. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"saddle_stationary","scale":16}。請依 [mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(saddle_stationary, f(x,y)=16*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)

Wave6第七層核對=(saddle_stationary, f(x,y)=16*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0091/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 92. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"incomplete_coordinate_list","scale":17}。請依 [mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(incomplete_coordinate_list, 3, [17*partial_x, 17*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)

Wave6第七層核對=(incomplete_coordinate_list, 3, [17*partial_x, 17*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0092/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 93. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"nondifferentiable_norm","scale":18}。請依 [mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(nondifferentiable_norm, f(x,y)=18*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)

Wave6第七層核對=(nondifferentiable_norm, f(x,y)=18*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0093/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 94. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"partials_exist_not_differentiable","scale":19}。請依 [mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(partials_exist_not_differentiable, f(x,y)=19*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [19, 0], 是, along_y=x:f(x,x)=19/2*x_but_linear_candidate=19*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)

Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=19*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [19, 0], 是, along_y=x:f(x,x)=19/2*x_but_linear_candidate=19*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0094/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 95. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"saddle_stationary","scale":20}。請依 [mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)

Wave6第七層核對=(saddle_stationary, f(x,y)=20*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0095/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 96. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"incomplete_coordinate_list","scale":21}。請依 [mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(incomplete_coordinate_list, 3, [21*partial_x, 21*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)

Wave6第七層核對=(incomplete_coordinate_list, 3, [21*partial_x, 21*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0096/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 97. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"nondifferentiable_norm","scale":22}。請依 [mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(nondifferentiable_norm, f(x,y)=22*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)

Wave6第七層核對=(nondifferentiable_norm, f(x,y)=22*sqrt(x^2+y^2), [0, 0], 否, 否, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0097/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 98. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"partials_exist_not_differentiable","scale":23}。請依 [mode, function_description, point, coordinate_partial_values, coordinate_partials_exist, diagonal_comparison, differentiable, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(partials_exist_not_differentiable, f(x,y)=23*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [23, 0], 是, along_y=x:f(x,x)=23/2*x_but_linear_candidate=23*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)

Wave6第七層核對=(partials_exist_not_differentiable, f(x,y)=23*x^3/(x^2+y^2)_off_origin;f(0,0)=0, [0, 0], [23, 0], 是, along_y=x:f(x,x)=23/2*x_but_linear_candidate=23*x, 否, both_coordinate_partials_exist_but_the_diagonal_remainder_is_not_little_o_of_the_norm)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0098/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 99. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"saddle_stationary","scale":24}。請依 [mode, function_description, point, gradient_values, stationary, extremum_certificate, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(saddle_stationary, f(x,y)=24*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)

Wave6第七層核對=(saddle_stationary, f(x,y)=24*(x^2-y^2), [0, 0], [0, 0], 是, 否, zero_gradient_at_a_saddle_is_not_an_extremum_certificate)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0099/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate





## 100. 梯度向量

standard Euclidean coordinates中的完整ordered partial-derivative vector。操作=gradient-boundary-counterexample；資料={"mode":"incomplete_coordinate_list","scale":25}。請依 [mode, required_dimension, provided_component_descriptions, provided_count, complete_gradient, boundary] 欄序回答。

mode選6、8或7欄；partials_exist_not_differentiable的向量是coordinate partial values，不因兩偏導存在就證total differentiability。不補欄、不從答案猜mode。

查看答案與解釋(incomplete_coordinate_list, 3, [25*partial_x, 25*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)

Wave6第七層核對=(incomplete_coordinate_list, 3, [25*partial_x, 25*partial_y], 2, 否, missing_coordinate_component_is_not_a_gradient_in_R3)；依完整明示資料獨立重算；polynomial symbolic gradient與rational point evaluation使用exact arithmetic；partial存在不一般推出differentiability；gradient zero只證stationary，不證extremum或global descent。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261113-r06-gradient-vector-0100/)

梯度向量 · qf-mod44-gradient-boundary-counterexample · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261113-r06-gradient-vector-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261113-r06-gradient-vector-foundations/validation-report.json)