# Wave6 Thirteenth Layer: Newton, Trapezoids, Gradient Descent, Lagrange Conditions and Numerical-Optimization Depth Complete four new traceable knowledge objects and deepen three existing objects in dependency order on the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Real derivatives at a point and finite difference quotients](/en/mko/mko-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-derivative) [Concept guides](/en/derivations/?q=mko-derivative) - Stage 1 · required ## [Algorithm correctness](/en/mko/mko-algorithm-correctness/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-algorithm-correctness) [Concept guides](/en/derivations/?q=mko-algorithm-correctness) - Stage 1 · required ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-equation) [Concept guides](/en/derivations/?q=mko-equation) - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Differentiating polynomial sums, products, and compositions](/en/mko/mko-derivative-rules/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-derivative-rules) [Concept guides](/en/derivations/?q=mko-derivative-rules) - Stage 1 · required ## [Signed polynomial definite integrals on finite intervals](/en/mko/mko-definite-integral/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-definite-integral) [Concept guides](/en/derivations/?q=mko-definite-integral) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 1 · required ## [Real vector space](/en/mko/mko-real-vector-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-vector-space) [Concept guides](/en/derivations/?q=mko-real-vector-space) - Stage 1 · required ## [Linear systems in two variables and degenerate constraints](/en/mko/mko-system-linear-equations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-system-linear-equations) [Concept guides](/en/derivations/?q=mko-system-linear-equations) - Stage 1 · required ## [Matrix rank](/en/mko/mko-matrix-rank/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-matrix-rank) [Concept guides](/en/derivations/?q=mko-matrix-rank) - Stage 1 · required ## [Linear independence and bases](/en/mko/mko-linear-independence-basis/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-independence-basis) [Concept guides](/en/derivations/?q=mko-linear-independence-basis) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value) - Stage 1 · required ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-ratio-proportion) [Concept guides](/en/derivations/?q=mko-ratio-proportion) - Stage 1 · required ## [Partial derivative](/en/mko/mko-partial-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-partial-derivative) [Concept guides](/en/derivations/?q=mko-partial-derivative) - Stage 1 · required ## [Plane vectors in a fixed basis](/en/mko/mko-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-vector) [Concept guides](/en/derivations/?q=mko-vector) - Stage 1 · required ## [Multivariable real-valued function](/en/mko/mko-multivariable-real-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-multivariable-real-function) [Concept guides](/en/derivations/?q=mko-multivariable-real-function) - Stage 1 · required ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 2 · optional ## [Numerical approximation and error](/en/mko/mko-numerical-approximation-error/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Extend the existing error object with affine residual versus forward error, propagation, conditioning witnesses and tolerance boundaries. [Related questions](/en/explore/?mko=mko-numerical-approximation-error) [Concept guides](/en/derivations/?q=mko-numerical-approximation-error) - Stage 2 · optional ## [Gradient vector](/en/mko/mko-gradient-vector/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Extend fixed standard Euclidean semantics with gradient linearity, negative-gradient inner products, symmetric-quadratic differences and metric boundaries. [Related questions](/en/explore/?mko=mko-gradient-vector) [Concept guides](/en/derivations/?q=mko-gradient-vector) - Stage 2 · optional ## [Optimization problem](/en/mko/mko-optimization-problem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) Separate finite neighborhood-local and global minima, strict improvement, attained lower bounds and empty-domain boundaries on the complete feasible structure. [Related questions](/en/explore/?mko=mko-optimization-problem) [Concept guides](/en/derivations/?q=mko-optimization-problem) - Stage 3 · optional ## [Newton method for root finding](/en/mko/mko-newton-method/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) At a qualified point with nonzero derivative, Newton method moves to the tangent-line zero x−f(x)/f_prime(x); convergence requires separate local hypotheses. [Related questions](/en/explore/?mko=mko-newton-method) [Concept guides](/en/derivations/?q=mko-newton-method) - Stage 3 · optional ## [Composite trapezoidal quadrature](/en/mko/mko-trapezoidal-quadrature/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) On a strictly increasing partition, add each subinterval width times its two-endpoint average to approximate the definite integral. [Related questions](/en/explore/?mko=mko-trapezoidal-quadrature) [Concept guides](/en/derivations/?q=mko-trapezoidal-quadrature) - Stage 3 · optional ## [Gradient descent](/en/mko/mko-gradient-descent/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) In fixed Euclidean coordinates, a positive step moves opposite the gradient; descent and convergence still require objective and step-size hypotheses. [Related questions](/en/explore/?mko=mko-gradient-descent) [Concept guides](/en/derivations/?q=mko-gradient-descent) - Stage 3 · optional ## [Necessary Lagrange multiplier condition for equality constraints](/en/mko/mko-lagrange-multiplier-condition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261119%2F) At a regular feasible constrained local extremum, the objective gradient is a linear combination of the constraint gradients. [Related questions](/en/explore/?mko=mko-lagrange-multiplier-condition) [Concept guides](/en/derivations/?q=mko-lagrange-multiplier-condition) Path source & statuspath-pulse-foundations-20261119 · candidate [Original paths JSON](/data/architecture/learning-paths.json)