# 梯度密度：線性、負梯度方向與二次差分基礎 · 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-20261119-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

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[1,3],"gradient_f":[1,0],"gradient_g":[0,-1]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([1, 0], [0, -1], 1/2, 1/3, [1/2, -1/3], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([1, 0], [0, -1], 1/2, 1/3, [1/2, -1/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0001/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 2. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[2,3],"gradient_f":[2,1],"gradient_g":[1,-2]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([2, 1], [1, -2], 1/2, 2/3, [5/3, -5/6], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([2, 1], [1, -2], 1/2, 2/3, [5/3, -5/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0002/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 3. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[3,3],"gradient_f":[3,2],"gradient_g":[2,-3]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([3, 2], [2, -3], 1/2, 1, [7/2, -2], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([3, 2], [2, -3], 1/2, 1, [7/2, -2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0003/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 4. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[4,3],"gradient_f":[4,3],"gradient_g":[3,-4]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([4, 3], [3, -4], 1/2, 4/3, [6, -23/6], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([4, 3], [3, -4], 1/2, 4/3, [6, -23/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0004/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 5. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[1,3],"gradient_f":[5,4],"gradient_g":[4,-5]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([5, 4], [4, -5], 1/2, 1/3, [23/6, 1/3], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([5, 4], [4, -5], 1/2, 1/3, [23/6, 1/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0005/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 6. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[2,3],"gradient_f":[6,0],"gradient_g":[5,-6]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([6, 0], [5, -6], 1/2, 2/3, [19/3, -4], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([6, 0], [5, -6], 1/2, 2/3, [19/3, -4], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0006/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 7. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[3,3],"gradient_f":[7,1],"gradient_g":[6,-7]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([7, 1], [6, -7], 1/2, 1, [19/2, -13/2], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([7, 1], [6, -7], 1/2, 1, [19/2, -13/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0007/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 8. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[4,3],"gradient_f":[8,2],"gradient_g":[0,-8]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([8, 2], [0, -8], 1/2, 4/3, [4, -29/3], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([8, 2], [0, -8], 1/2, 4/3, [4, -29/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0008/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 9. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[1,3],"gradient_f":[9,3],"gradient_g":[1,-9]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([9, 3], [1, -9], 1/2, 1/3, [29/6, -3/2], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([9, 3], [1, -9], 1/2, 1/3, [29/6, -3/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0009/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 10. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[2,3],"gradient_f":[10,4],"gradient_g":[2,-10]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([10, 4], [2, -10], 1/2, 2/3, [19/3, -14/3], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([10, 4], [2, -10], 1/2, 2/3, [19/3, -14/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0010/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 11. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[3,3],"gradient_f":[11,0],"gradient_g":[3,-11]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([11, 0], [3, -11], 1/2, 1, [17/2, -11], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([11, 0], [3, -11], 1/2, 1, [17/2, -11], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0011/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 12. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[4,3],"gradient_f":[12,1],"gradient_g":[4,-12]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([12, 1], [4, -12], 1/2, 4/3, [34/3, -31/2], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([12, 1], [4, -12], 1/2, 4/3, [34/3, -31/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0012/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 13. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[1,3],"gradient_f":[13,2],"gradient_g":[5,-13]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([13, 2], [5, -13], 1/2, 1/3, [49/6, -10/3], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([13, 2], [5, -13], 1/2, 1/3, [49/6, -10/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0013/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 14. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[2,3],"gradient_f":[14,3],"gradient_g":[6,-14]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([14, 3], [6, -14], 1/2, 2/3, [11, -47/6], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([14, 3], [6, -14], 1/2, 2/3, [11, -47/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0014/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 15. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[3,3],"gradient_f":[15,4],"gradient_g":[0,-15]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([15, 4], [0, -15], 1/2, 1, [15/2, -13], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([15, 4], [0, -15], 1/2, 1, [15/2, -13], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0015/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 16. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[4,3],"gradient_f":[16,0],"gradient_g":[1,-16]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([16, 0], [1, -16], 1/2, 4/3, [28/3, -64/3], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([16, 0], [1, -16], 1/2, 4/3, [28/3, -64/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0016/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 17. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[1,3],"gradient_f":[17,1],"gradient_g":[2,-17]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([17, 1], [2, -17], 1/2, 1/3, [55/6, -31/6], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([17, 1], [2, -17], 1/2, 1/3, [55/6, -31/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0017/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 18. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[2,3],"gradient_f":[18,2],"gradient_g":[3,-18]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([18, 2], [3, -18], 1/2, 2/3, [11, -11], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([18, 2], [3, -18], 1/2, 2/3, [11, -11], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0018/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 19. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[3,3],"gradient_f":[19,3],"gradient_g":[4,-19]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([19, 3], [4, -19], 1/2, 1, [27/2, -35/2], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([19, 3], [4, -19], 1/2, 1, [27/2, -35/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0019/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 20. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[4,3],"gradient_f":[20,4],"gradient_g":[5,-20]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([20, 4], [5, -20], 1/2, 4/3, [50/3, -74/3], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([20, 4], [5, -20], 1/2, 4/3, [50/3, -74/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0020/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 21. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[1,3],"gradient_f":[21,0],"gradient_g":[6,-21]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([21, 0], [6, -21], 1/2, 1/3, [25/2, -7], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([21, 0], [6, -21], 1/2, 1/3, [25/2, -7], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0021/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 22. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[2,3],"gradient_f":[22,1],"gradient_g":[0,-22]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([22, 1], [0, -22], 1/2, 2/3, [11, -85/6], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([22, 1], [0, -22], 1/2, 2/3, [11, -85/6], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0022/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 23. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[3,3],"gradient_f":[23,2],"gradient_g":[1,-23]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([23, 2], [1, -23], 1/2, 1, [25/2, -22], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([23, 2], [1, -23], 1/2, 1, [25/2, -22], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0023/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 24. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[4,3],"gradient_f":[24,3],"gradient_g":[2,-24]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([24, 3], [2, -24], 1/2, 4/3, [44/3, -61/2], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([24, 3], [2, -24], 1/2, 4/3, [44/3, -61/2], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0024/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 25. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-linearity；明示資料={"alpha":[1,2],"beta":[1,3],"gradient_f":[25,4],"gradient_g":[3,-25]}；請依 [gradient_f, gradient_g, alpha, beta, combined_gradient, coordinate_scope] 回答。

查看答案與解釋([25, 4], [3, -25], 1/2, 1/3, [27/2, -19/3], pointwise_gradient_linearity_in_fixed_coordinates)

Wave6第十三層核對=([25, 4], [3, -25], 1/2, 1/3, [27/2, -19/3], pointwise_gradient_linearity_in_fixed_coordinates)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0025/)

梯度向量 · qf-mod50-graddeep-linearity · candidate





## 26. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[1,-1]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([1, -1], [-1, 1], -2, 2, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([1, -1], [-1, 1], -2, 2, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0026/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 27. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[2,-2]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([2, -2], [-2, 2], -8, 8, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([2, -2], [-2, 2], -8, 8, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0027/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 28. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[3,-3]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([3, -3], [-3, 3], -18, 18, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([3, -3], [-3, 3], -18, 18, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0028/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 29. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[4,-4]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([4, -4], [-4, 4], -32, 32, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([4, -4], [-4, 4], -32, 32, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0029/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 30. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[5,-5]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([5, -5], [-5, 5], -50, 50, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([5, -5], [-5, 5], -50, 50, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0030/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 31. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[6,-6]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([6, -6], [-6, 6], -72, 72, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([6, -6], [-6, 6], -72, 72, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0031/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 32. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[7,-7]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([7, -7], [-7, 7], -98, 98, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([7, -7], [-7, 7], -98, 98, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0032/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 33. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[8,-8]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([8, -8], [-8, 8], -128, 128, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([8, -8], [-8, 8], -128, 128, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0033/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 34. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[9,-9]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([9, -9], [-9, 9], -162, 162, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([9, -9], [-9, 9], -162, 162, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0034/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 35. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[10,-10]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([10, -10], [-10, 10], -200, 200, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([10, -10], [-10, 10], -200, 200, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0035/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 36. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[11,-11]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([11, -11], [-11, 11], -242, 242, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([11, -11], [-11, 11], -242, 242, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0036/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 37. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[12,-12]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([12, -12], [-12, 12], -288, 288, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([12, -12], [-12, 12], -288, 288, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0037/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 38. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[13,-13]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([13, -13], [-13, 13], -338, 338, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([13, -13], [-13, 13], -338, 338, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0038/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 39. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[14,-14]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([14, -14], [-14, 14], -392, 392, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([14, -14], [-14, 14], -392, 392, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0039/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 40. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[15,-15]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([15, -15], [-15, 15], -450, 450, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([15, -15], [-15, 15], -450, 450, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0040/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 41. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[16,-16]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([16, -16], [-16, 16], -512, 512, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([16, -16], [-16, 16], -512, 512, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0041/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 42. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[17,-17]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([17, -17], [-17, 17], -578, 578, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([17, -17], [-17, 17], -578, 578, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0042/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 43. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[18,-18]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([18, -18], [-18, 18], -648, 648, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([18, -18], [-18, 18], -648, 648, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0043/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 44. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[19,-19]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([19, -19], [-19, 19], -722, 722, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([19, -19], [-19, 19], -722, 722, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0044/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 45. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[20,-20]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([20, -20], [-20, 20], -800, 800, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([20, -20], [-20, 20], -800, 800, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0045/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 46. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[21,-21]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([21, -21], [-21, 21], -882, 882, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([21, -21], [-21, 21], -882, 882, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0046/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 47. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[22,-22]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([22, -22], [-22, 22], -968, 968, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([22, -22], [-22, 22], -968, 968, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0047/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 48. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[23,-23]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([23, -23], [-23, 23], -1058, 1058, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([23, -23], [-23, 23], -1058, 1058, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0048/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 49. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[24,-24]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([24, -24], [-24, 24], -1152, 1152, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([24, -24], [-24, 24], -1152, 1152, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0049/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 50. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-negative-direction；明示資料={"gradient":[25,-25]}；請依 [gradient, negative_gradient, directional_inner_product, gradient_norm_squared, strictly_negative, first_order_scope] 回答。

查看答案與解釋([25, -25], [-25, 25], -1250, 1250, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)

Wave6第十三層核對=([25, -25], [-25, 25], -1250, 1250, 是, negative_gradient_has_strictly_negative_directional_inner_product_at_a_nonstationary_point)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0050/)

梯度向量 · qf-mod50-graddeep-negative-direction · candidate





## 51. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[1,1],[1,2]],"x":[0,1],"y":[1,0]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

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

Wave6第十三層核對=([[1, 1], [1, 2]], [0, 1], [1, 0], [-1, 1], [0, 1], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0051/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 52. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[2,1],[1,3]],"x":[1,1],"y":[1,-1]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

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

Wave6第十三層核對=([[2, 1], [1, 3]], [1, 1], [1, -1], [0, 2], [2, 6], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0052/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 53. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[3,1],[1,4]],"x":[2,1],"y":[1,-2]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[3, 1], [1, 4]], [2, 1], [1, -2], [1, 3], [6, 13], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[3, 1], [1, 4]], [2, 1], [1, -2], [1, 3], [6, 13], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0053/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 54. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[4,1],[1,5]],"x":[3,1],"y":[1,-3]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[4, 1], [1, 5]], [3, 1], [1, -3], [2, 4], [12, 22], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[4, 1], [1, 5]], [3, 1], [1, -3], [2, 4], [12, 22], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0054/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 55. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[5,1],[1,6]],"x":[4,1],"y":[1,-4]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[5, 1], [1, 6]], [4, 1], [1, -4], [3, 5], [20, 33], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[5, 1], [1, 6]], [4, 1], [1, -4], [3, 5], [20, 33], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0055/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 56. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[6,1],[1,2]],"x":[5,1],"y":[1,-5]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[6, 1], [1, 2]], [5, 1], [1, -5], [4, 6], [30, 16], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[6, 1], [1, 2]], [5, 1], [1, -5], [4, 6], [30, 16], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0056/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 57. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[7,1],[1,3]],"x":[6,1],"y":[1,-6]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[7, 1], [1, 3]], [6, 1], [1, -6], [5, 7], [42, 26], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[7, 1], [1, 3]], [6, 1], [1, -6], [5, 7], [42, 26], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0057/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 58. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[8,1],[1,4]],"x":[7,1],"y":[1,-7]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[8, 1], [1, 4]], [7, 1], [1, -7], [6, 8], [56, 38], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[8, 1], [1, 4]], [7, 1], [1, -7], [6, 8], [56, 38], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0058/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 59. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[9,1],[1,5]],"x":[8,1],"y":[1,-8]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[9, 1], [1, 5]], [8, 1], [1, -8], [7, 9], [72, 52], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[9, 1], [1, 5]], [8, 1], [1, -8], [7, 9], [72, 52], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0059/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 60. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[10,1],[1,6]],"x":[9,1],"y":[1,-9]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[10, 1], [1, 6]], [9, 1], [1, -9], [8, 10], [90, 68], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[10, 1], [1, 6]], [9, 1], [1, -9], [8, 10], [90, 68], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0060/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 61. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[11,1],[1,2]],"x":[10,1],"y":[1,-10]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[11, 1], [1, 2]], [10, 1], [1, -10], [9, 11], [110, 31], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[11, 1], [1, 2]], [10, 1], [1, -10], [9, 11], [110, 31], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0061/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 62. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[12,1],[1,3]],"x":[11,1],"y":[1,-11]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[12, 1], [1, 3]], [11, 1], [1, -11], [10, 12], [132, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[12, 1], [1, 3]], [11, 1], [1, -11], [10, 12], [132, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0062/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 63. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[13,1],[1,4]],"x":[12,1],"y":[1,-12]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[13, 1], [1, 4]], [12, 1], [1, -12], [11, 13], [156, 63], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[13, 1], [1, 4]], [12, 1], [1, -12], [11, 13], [156, 63], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0063/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 64. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[14,1],[1,5]],"x":[13,1],"y":[1,-13]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[14, 1], [1, 5]], [13, 1], [1, -13], [12, 14], [182, 82], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[14, 1], [1, 5]], [13, 1], [1, -13], [12, 14], [182, 82], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0064/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 65. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[15,1],[1,6]],"x":[14,1],"y":[1,-14]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[15, 1], [1, 6]], [14, 1], [1, -14], [13, 15], [210, 103], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[15, 1], [1, 6]], [14, 1], [1, -14], [13, 15], [210, 103], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0065/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 66. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[16,1],[1,2]],"x":[15,1],"y":[1,-15]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[16, 1], [1, 2]], [15, 1], [1, -15], [14, 16], [240, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[16, 1], [1, 2]], [15, 1], [1, -15], [14, 16], [240, 46], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0066/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 67. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[17,1],[1,3]],"x":[16,1],"y":[1,-16]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[17, 1], [1, 3]], [16, 1], [1, -16], [15, 17], [272, 66], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[17, 1], [1, 3]], [16, 1], [1, -16], [15, 17], [272, 66], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0067/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 68. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[18,1],[1,4]],"x":[17,1],"y":[1,-17]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[18, 1], [1, 4]], [17, 1], [1, -17], [16, 18], [306, 88], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[18, 1], [1, 4]], [17, 1], [1, -17], [16, 18], [306, 88], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0068/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 69. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[19,1],[1,5]],"x":[18,1],"y":[1,-18]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[19, 1], [1, 5]], [18, 1], [1, -18], [17, 19], [342, 112], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[19, 1], [1, 5]], [18, 1], [1, -18], [17, 19], [342, 112], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0069/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 70. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[20,1],[1,6]],"x":[19,1],"y":[1,-19]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[20, 1], [1, 6]], [19, 1], [1, -19], [18, 20], [380, 138], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[20, 1], [1, 6]], [19, 1], [1, -19], [18, 20], [380, 138], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0070/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 71. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[21,1],[1,2]],"x":[20,1],"y":[1,-20]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[21, 1], [1, 2]], [20, 1], [1, -20], [19, 21], [420, 61], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[21, 1], [1, 2]], [20, 1], [1, -20], [19, 21], [420, 61], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0071/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 72. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[22,1],[1,3]],"x":[21,1],"y":[1,-21]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[22, 1], [1, 3]], [21, 1], [1, -21], [20, 22], [462, 86], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[22, 1], [1, 3]], [21, 1], [1, -21], [20, 22], [462, 86], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0072/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 73. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[23,1],[1,4]],"x":[22,1],"y":[1,-22]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[23, 1], [1, 4]], [22, 1], [1, -22], [21, 23], [506, 113], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[23, 1], [1, 4]], [22, 1], [1, -22], [21, 23], [506, 113], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0073/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 74. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[24,1],[1,5]],"x":[23,1],"y":[1,-23]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[24, 1], [1, 5]], [23, 1], [1, -23], [22, 24], [552, 142], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[24, 1], [1, 5]], [23, 1], [1, -23], [22, 24], [552, 142], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0074/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 75. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-quadratic-difference；明示資料={"symmetric_matrix":[[25,1],[1,6]],"x":[24,1],"y":[1,-24]}；請依 [symmetric_matrix, x, y, point_difference, gradient_difference, quadratic_identity] 回答。

查看答案與解釋([[25, 1], [1, 6]], [24, 1], [1, -24], [23, 25], [600, 173], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)

Wave6第十三層核對=([[25, 1], [1, 6]], [24, 1], [1, -24], [23, 25], [600, 173], grad_f_x_minus_grad_f_y_equals_Q_times_x_minus_y_for_symmetric_quadratic)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0075/)

梯度向量 · qf-mod50-graddeep-quadratic-difference · candidate





## 76. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"zero_gradient","model":{"dimension":1}}；請依 [mode, dimension, zero_gradient, stationary, minimum_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(zero_gradient, 1, [0], 是, 否, zero_gradient_only_certifies_stationarity)

Wave6第十三層核對=(zero_gradient, 1, [0], 是, 否, zero_gradient_only_certifies_stationarity)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0076/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 77. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"positive_gradient_direction","model":{"gradient":[2,-1]}}；請依 [mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(positive_gradient_direction, [2, -1], [2, -1], 5, 是, 否, positive_gradient_is_first_order_ascent_not_descent)

Wave6第十三層核對=(positive_gradient_direction, [2, -1], [2, -1], 5, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0077/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 78. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"non_euclidean_metric","model":{"coordinate_gradient":[3,1],"metric":[[2,0],[0,3]]}}；請依 [mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(non_euclidean_metric, [3, 1], [[2, 0], [0, 3]], [3/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)

Wave6第十三層核對=(non_euclidean_metric, [3, 1], [[2, 0], [0, 3]], [3/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0078/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 79. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":4,"step":[1,5]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 4, 1/5, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 4, 1/5, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0079/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 80. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":5,"step":[1,6]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 5, 1/6, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 5, 1/6, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0080/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 81. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"positive_gradient_direction","model":{"gradient":[6,-1]}}；請依 [mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(positive_gradient_direction, [6, -1], [6, -1], 37, 是, 否, positive_gradient_is_first_order_ascent_not_descent)

Wave6第十三層核對=(positive_gradient_direction, [6, -1], [6, -1], 37, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0081/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 82. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"non_euclidean_metric","model":{"coordinate_gradient":[7,1],"metric":[[2,0],[0,3]]}}；請依 [mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(non_euclidean_metric, [7, 1], [[2, 0], [0, 3]], [7/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)

Wave6第十三層核對=(non_euclidean_metric, [7, 1], [[2, 0], [0, 3]], [7/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0082/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 83. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":8,"step":[1,9]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 8, 1/9, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 8, 1/9, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0083/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 84. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":9,"step":[1,10]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 9, 1/10, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 9, 1/10, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0084/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 85. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"positive_gradient_direction","model":{"gradient":[10,-1]}}；請依 [mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(positive_gradient_direction, [10, -1], [10, -1], 101, 是, 否, positive_gradient_is_first_order_ascent_not_descent)

Wave6第十三層核對=(positive_gradient_direction, [10, -1], [10, -1], 101, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0085/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 86. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"non_euclidean_metric","model":{"coordinate_gradient":[11,1],"metric":[[2,0],[0,3]]}}；請依 [mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(non_euclidean_metric, [11, 1], [[2, 0], [0, 3]], [11/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)

Wave6第十三層核對=(non_euclidean_metric, [11, 1], [[2, 0], [0, 3]], [11/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0086/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 87. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":12,"step":[1,13]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 12, 1/13, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 12, 1/13, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0087/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 88. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":13,"step":[1,14]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 13, 1/14, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 13, 1/14, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0088/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 89. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"positive_gradient_direction","model":{"gradient":[14,-1]}}；請依 [mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(positive_gradient_direction, [14, -1], [14, -1], 197, 是, 否, positive_gradient_is_first_order_ascent_not_descent)

Wave6第十三層核對=(positive_gradient_direction, [14, -1], [14, -1], 197, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0089/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 90. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"non_euclidean_metric","model":{"coordinate_gradient":[15,1],"metric":[[2,0],[0,3]]}}；請依 [mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(non_euclidean_metric, [15, 1], [[2, 0], [0, 3]], [15/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)

Wave6第十三層核對=(non_euclidean_metric, [15, 1], [[2, 0], [0, 3]], [15/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0090/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 91. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":16,"step":[1,17]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 16, 1/17, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 16, 1/17, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0091/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 92. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":17,"step":[1,18]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 17, 1/18, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 17, 1/18, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0092/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 93. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"positive_gradient_direction","model":{"gradient":[18,-1]}}；請依 [mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(positive_gradient_direction, [18, -1], [18, -1], 325, 是, 否, positive_gradient_is_first_order_ascent_not_descent)

Wave6第十三層核對=(positive_gradient_direction, [18, -1], [18, -1], 325, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0093/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 94. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"non_euclidean_metric","model":{"coordinate_gradient":[19,1],"metric":[[2,0],[0,3]]}}；請依 [mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(non_euclidean_metric, [19, 1], [[2, 0], [0, 3]], [19/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)

Wave6第十三層核對=(non_euclidean_metric, [19, 1], [[2, 0], [0, 3]], [19/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0094/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 95. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":20,"step":[1,21]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 20, 1/21, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 20, 1/21, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0095/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 96. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":21,"step":[1,22]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 21, 1/22, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 21, 1/22, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0096/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 97. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"positive_gradient_direction","model":{"gradient":[22,-1]}}；請依 [mode, gradient, positive_direction, directional_inner_product, positive, descent_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(positive_gradient_direction, [22, -1], [22, -1], 485, 是, 否, positive_gradient_is_first_order_ascent_not_descent)

Wave6第十三層核對=(positive_gradient_direction, [22, -1], [22, -1], 485, 是, 否, positive_gradient_is_first_order_ascent_not_descent)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0097/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 98. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"non_euclidean_metric","model":{"coordinate_gradient":[23,1],"metric":[[2,0],[0,3]]}}；請依 [mode, coordinate_partials_covector, positive_definite_metric, metric_gradient, metric_independence_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(non_euclidean_metric, [23, 1], [[2, 0], [0, 3]], [23/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)

Wave6第十三層核對=(non_euclidean_metric, [23, 1], [[2, 0], [0, 3]], [23/2, 1/3], 否, coordinate_partials_become_a_gradient_vector_only_after_the_metric_identification)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0098/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 99. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":24,"step":[1,25]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 24, 1/25, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 24, 1/25, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0099/)

梯度向量 · qf-mod50-graddeep-boundary · candidate





## 100. 梯度向量

fixed standard Euclidean coordinates中的完整ordered gradient vector。操作=graddeep-boundary；明示資料={"mode":"finite_difference_confusion","model":{"difference_quotient":25,"step":[1,26]}}；請依 [mode, finite_difference_quotient, nonzero_step, derivative_or_gradient_certificate_claim, boundary] 回答。

四種mode為6/7/6/5欄。非Euclidean metric的gradient須由metric identification取得，不可直接等同coordinate partial covector。finite difference不是derivative證書。

查看答案與解釋(finite_difference_confusion, 25, 1/26, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)

Wave6第十三層核對=(finite_difference_confusion, 25, 1/26, 否, one_finite_difference_quotient_is_not_a_derivative_or_gradient_certificate)；依完整明示資料精確重算；gradient線性、negative-gradient inner product與symmetric quadratic difference皆以exact arithmetic核對；zero gradient只證stationary；metric改變需重新identify covector，finite difference不等於gradient。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r06-0100/)

梯度向量 · qf-mod50-graddeep-boundary · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r06-gradient-vector-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261119-r06-gradient-vector-foundations/validation-report.json)