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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
Threads
Multivariable calculus
Vector algebra
Math 2374
Calculus Refresher
Interactive Gallery of Quadric Surfaces
University of Minnesota, Math 1241
University of Portland, Math 201
Elementary dynamical systems
Network tutorial
Elementary math
Advanced elementary math