Advanced Multivariable Calculus

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• Norms and distance in Rⁿ ~8 min
• Open sets, closed sets, and boundaries ~10 min
• Interior, closure, and limit points ~8 min
• Sequences in Rⁿ and convergence ~8 min
• Compactness — closed and bounded ~10 min
• The Extreme Value Theorem in n dimensions ~8 min
• Connected and path-connected sets ~8 min
• Simply connected versus multiply connected regions ~8 min
• Uniform continuity and why it matters ~8 min

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• From partial derivatives to the total derivative ~10 min
• The derivative as the best linear approximation ~12 min
• The Jacobian matrix ~10 min
• Continuity, partials, and differentiability — what implies what ~10 min
• The C¹ theorem ~8 min
• The chain rule as matrix multiplication ~10 min
• Directional derivatives as the derivative applied to a vector ~8 min
• The Hessian as a bilinear form ~12 min
• Multi-index notation and Taylor's theorem in n variables ~10 min

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• Local invertibility and the question it answers ~8 min
• The Inverse Function Theorem ~12 min
• Reading the IFT geometrically ~8 min
• The Implicit Function Theorem ~12 min
• Solving for some variables in terms of others ~8 min
• Constraint surfaces, tangent spaces, and level sets ~10 min
• The constant rank theorem ~10 min
• Lagrange multipliers, justified ~10 min
• Second-order conditions and the bordered Hessian ~12 min

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• Coordinate systems as maps ~8 min
• Coordinate curves, surfaces, and local basis vectors ~8 min
• Scale factors and the line element ~10 min
• Orthogonal curvilinear coordinates ~8 min
• The gradient in curvilinear coordinates ~8 min
• The divergence in curvilinear coordinates ~10 min
• The curl in curvilinear coordinates ~10 min
• The Laplacian in curvilinear coordinates ~8 min
• Beyond cylindrical and spherical ~8 min

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• The Einstein summation convention ~8 min
• The Kronecker delta ~6 min
• The Levi-Civita symbol ~10 min
• The epsilon-delta identity and vector identities made mechanical ~12 min
• What a tensor is — transformation under rotation ~10 min
• Second-order tensors — inertia, stress, strain ~8 min
• Symmetric and antisymmetric parts ~8 min
• Principal axes and eigen-decomposition ~10 min
• The del operator in index form ~8 min
• The word "tensor" in machine learning ~6 min

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• Why forms — the trouble with dx dy ~8 min
• 1-forms and the differential of a function ~8 min
• The wedge product ~10 min
• 2-forms and 3-forms in R³ ~8 min
• k-forms in Rⁿ and the dimension count ~8 min
• The exterior derivative ~10 min
• d² = 0 and what it encodes ~8 min
• Grad, curl, and div are all d ~10 min
• The Hodge star and why R³ is special ~8 min
• Pullbacks and change of variables ~10 min

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• Orientation, made precise ~8 min
• Chains, cells, and the boundary operator ~10 min
• ∂² = 0 and the boundary of a boundary ~8 min
• Integrating a k-form over a k-chain ~8 min
• The generalized Stokes theorem ~12 min
• Recovering the FTC, Green's, Stokes', and the Divergence Theorem ~10 min
• Closed forms and exact forms ~8 min
• The Poincaré lemma ~10 min
• De Rham cohomology — when closed isn't exact ~12 min

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• Jordan measure and content zero ~8 min
• The Riemann integral in Rⁿ, done properly ~8 min
• When Fubini's theorem fails ~10 min
• Improper multiple integrals and absolute convergence ~8 min
• The Gaussian integral ~10 min
• Differentiation under the integral sign ~10 min
• The Leibniz rule with moving boundaries ~8 min
• The Gamma and Beta functions ~10 min
• The volume of the n-ball and high-dimensional geometry ~10 min

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• Harmonic functions and Laplace's equation ~8 min
• The mean value property ~10 min
• The maximum principle ~8 min
• Green's identities ~10 min
• Uniqueness for Dirichlet and Neumann problems ~8 min
• The Dirac delta and point sources ~8 min
• Green's functions and the fundamental solution ~10 min
• Solid angle and Gauss's law ~8 min
• The Helmholtz decomposition ~12 min
• Scalar and vector potentials, and gauge freedom ~8 min

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• The material derivative ~8 min
• The Reynolds transport theorem ~10 min
• Continuity equations and conservation laws ~8 min
• Maxwell's equations in integral form ~10 min
• Maxwell's equations in differential form ~10 min
• Maxwell's equations in the language of forms ~10 min
• The wave equation from Maxwell ~10 min
• Vorticity, circulation, and Kelvin's theorem ~10 min
• Divergence, diffusion, and the discrete Laplacian ~8 min

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• Functionals and the variational problem ~8 min
• The first variation ~10 min
• The Euler-Lagrange equation ~12 min
• The brachistochrone and the catenary ~10 min
• Several dependent variables ~8 min
• Several independent variables and field equations ~10 min
• Isoperimetric constraints and Lagrange multipliers for functionals ~10 min
• Geodesics and the metric ~10 min
• Noether's theorem — symmetry and conservation ~10 min
• Hamilton's principle and Lagrangian mechanics ~12 min