Multivariable Calculus Summary

GRE Math Subject · Axiom Academy

SUMMARY Multivariable Calculus Partial Derivatives: Differentiation with respect to one variable while treating others as constants Multiple Integrals: Double and triple integrals for computing volumes, areas, and other quantities Vector Fields: Functions that assign vectors to points in space Line Integrals: Integration along curves, including work integrals Surface Integrals: Integration over surfaces in 3D space Coordinate Transformations: Polar, cylindrical, and spherical coordinates with appropriate Jacobians Vector Calculus Theorems: Green's Theorem, Stokes' Theorem, Divergence Theorem Optimization: Lagrange multipliers for constrained extrema Green's Theorem: Relates a line integral around a closed curve to a double integral over the enclosed region. Fundamental for 2D vector calculus. Stokes' Theorem: The 3D analogue of Green's Theorem. Relates a line integral around the boundary of a surface to the surface integral of the curl. Divergence Theorem: Relates a surface integral over a closed surface to a volume integral over the enclosed region. Measures net flow out of a volume. Forgetting the Jacobian when changing coordinate systems Incorrect orientation (counterclockwise vs clockwise) in Green's Theorem Mixing up curl and divergence—curl applies to vector fields, divergence also applies to vector fields but measures different properties In Lagrange multipliers, not actually solving the constraint equation g(x,y) = 0

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