Thread: Multivariable calculus Covers basic pages in multivariable calculus Differential Calculus Partial derivatives Introduction to partial derivatives Partial derivative examples Partial derivative by limit definition Differentiability and the derivative Introduction to differentiability in higher dimensions The definition of differentiability in higher dimensions Subtleties of differentiability in higher dimensions The multivariable linear approximation The derivative matrix Examples of calculating the derivative The multidimensional differentiability theorem Non-differentiable functions must have discontinuous partial derivatives A differentiable function with discontinuous partial derivatives The chain rule Introduction to the multivariable chain rule Special cases of the multivariable chain rule Multivariable chain rule examples The directional derivative and the gradient An introduction to the directional derivative and the gradient Derivation of the directional derivative and the gradient Directional derivative and gradient examples Applications of differential calculus Introduction to Taylor's theorem for multivariable functions Multivariable Taylor polynomial example Introduction to local extrema of functions of two variables Two variable local extrema examples Integral calculus Double integrals Introduction to double integrals Double integrals as iterated integrals Double integral examples Double integrals as volume Double integrals as area Examples of changing the order of integration in double integrals Double integrals where one integration order is easier Triple Integrals Introduction to triple integrals The shadow method for determining triple integral bounds The cross section method for determining triple integral bounds Triple integral examples Changing variables Introduction to changing variables in double integrals Area calculation for changing variables in double integrals Double integral change of variable examples Illustrated example of changing variables in double integrals Triple integral change of variables story Introduction to changing variables in triple integrals Volume calculation for changing variables in triple integrals Triple integral change of variable examples Curves and surfaces Parametrized curves An introduction to parametrized curves Derivatives of parameterized curves Parametrized curve and derivative as location and velocity Tangent lines to parametrized curves Tangent line to parametrized curve examples Orienting curves Length of curves The arc length of a parametrized curve Parametrized curve arc length examples Parametrized surfaces An introduction to parametrized surfaces Parametrized surface examples Normal vector of parametrized surfaces Orienting surfaces A Möbius strip is not orientable Surface area Surface area of parametrized surfaces Calculation of the surface area of a parametrized surface Parametrized surface area example Vector fields Vector field basics Vector field overview Vector fields as fluid flow Vector operators The idea of the divergence of a vector field Subtleties about divergence The idea of the curl of a vector field Subtleties about curl The components of the curl Divergence and curl notation Divergence and curl example The definition of curl from line integrals Calculating the formula for circulation per unit area Integration over curves and surfaces Line integrals Introduction to a line integral of a scalar-valued function Introduction to a line integral of a vector field Line integrals are independent of parametrization Examples of scalar line integrals Alternate notation for vector line integrals Vector line integral examples Line integrals as circulation Surface integrals Introduction to a surface integral of a scalar-valued function Introduction to a surface integral of a vector field Scalar surface integral examples Vector surface integral examples Integration Synopsis The integrals of multivariable calculus Length, area, and volume factors The fundamental theorems of vector calculus Gradient theorem for line integrals An introduction to conservative vector fields The gradient theorem for line integrals A simple example of using the gradient theorem How to determine if a vector field is conservative A conservative vector field has no circulation A path-dependent vector field with zero curl Testing if three-dimensional vector fields are conservative Finding a potential function for conservative vector fields Finding a potential function for three-dimensional conservative vector fields Green's theorem The idea behind Green's theorem When Green's theorem applies Other ways of writing Green's theorem Green's theorem examples Using Green's theorem to find area Green's theorem with multiple boundary components Stokes' theorem The idea behind Stokes' theorem Proper orientation for Stokes' theorem Stokes' theorem examples Divergence theorem The idea behind the divergence theorem Divergence theorem examples Fundamental theorem synopsis The fundamental theorems of vector calculus