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Calculus

Differential calculus, limits, derivatives, chain rule backpropagation, multivariable partial derivatives, and optimization algorithms.

1Limits, Continuity & Differential Calculus Mechanics

Epsilon-delta limit definitions, derivative rate of change, power/product/chain rules, and gradient vectors.

Limits, Epsilon-Delta Rigor & Derivative Foundations

Formal definitions of limits, instantaneous rates of change, and automatic differentiation principles.

Epsilon-Delta Limit Definition, Derivatives & Difference Quotients

Mathematical formulation of limits via epsilon-delta bounds, derivative difference quotients, and numerical differentiation.

2Riemann Integration, Fundamental Theorem & Quadrature Mechanics

Darboux sums, Riemann integrability conditions, the Fundamental Theorem of Calculus, integration by parts, and adaptive numerical quadrature.

Riemann Sums, FTC Proofs & Adaptive Quadrature

Partition refinement, Darboux integral equality, FTC Part 1 and 2, and composite Simpson quadrature error bounds.

Riemann Integrability, the Fundamental Theorem of Calculus & Adaptive Simpson's Quadrature

Master upper and lower Darboux partitions, Lebesgue's integrability criterion, rigorous proofs of FTC parts 1 and 2, and the implementation of adaptive Simpson's quadrature.

3Multivariable Differential Calculus, Gradients & Hessians

Functions of several variables, partial derivatives, directional derivatives, gradient vectors, Jacobian matrices, and Hessian curvature matrices.

Multivariate Derivatives, Jacobian Matrices & Optimization

Directional derivatives, the gradient theorem, second-order Taylor expansions, Hessian eigenvalues, and multivariate optimization.

Multivariable Gradients, Hessian Matrices & Newton-Raphson Optimization

Master multivariable differentiability, the geometry of gradient vectors, Jacobian mappings, Hessian matrices of second partials, and multi-dimensional Newton-Raphson optimization.

4Vector Calculus, Differential Forms & Stokes' Theorems

Line integrals, surface integrals, conservative vector fields, Green's Theorem, Divergence Theorem, and Stokes' Generalized Theorem.

Flux, Circulation & Fundamental Integral Theorems

Path integrals, flux across closed surfaces, differential operators (curl, divergence), and the unification of Stokes' theorem.

Conservative Fields, Green's Theorem, the Divergence Theorem & Generalized Stokes Mechanics

Master path integrals, conservative force fields and potential functions, curl circulation density, divergence flux density, Green's Theorem, Gauss' Divergence Theorem, and Generalized Stokes formulation in exterior calculus.