Multivariable calculus
Categories: Calculus, Mathematics
Course Content
Thinking about multivariable functions

Introduction to multivariable calculus

Multivariable functions

Visualizing scalarvalued functions

Representing points in 3d

Introduction to 3d graphs

Interpreting graphs with slices

Contour plots

Visualizing vectorvalued functions

Parametric curves

Parametric surfaces

Vector fields, introduction

Fluid flow and vector fields

3d vector fields, introduction

3d vector field example

Transformations

Transformations, part 1

Transformations, part 2

Transformations, part 3
Derivatives of multivariable functions

Partial derivatives, introduction

Graphical understanding of partial derivatives

Formal definition of partial derivatives

Symmetry of second partial derivatives

Partial derivatives

Gradient

Gradient and graphs

Gradient and contour maps

Directional derivative

Directional derivative, formal definition

Directional derivatives and slope

Why the gradient is the direction of steepest ascent

Gradient and directional derivatives

Vectorvalued functions intro

Vectorvalued functions differentiation

Differential of a vector valued function

Vector valued function derivative example

Differentiating parametric curves

Multivariable chain rule

Multivariable chain rule intuition

Vector form of the multivariable chain rule

Multivariable chain rule and directional derivatives

More formal treatment of multivariable chain rule

Multivariable chain rule

Curvature intuition

Curvature formula, part 1

Curvature formula, part 2

Curvature formula, part 3

Curvature formula, part 4

Curvature formula, part 5

Curvature of a helix, part 1

Curvature of a helix, part 2

Curvature of a cycloid

Curvature

Computing the partial derivative of a vectorvalued function

Partial derivative of a parametric surface, part 1

Partial derivative of a parametric surface, part 2

Partial derivatives of vector fields

Partial derivatives of vector fields, component by component

Partial derivatives of vectorvalued functions

Divergence intuition, part 1

Divergence intuition, part 2

Divergence formula, part 1

Divergence formula, part 2

Divergence example

Divergence notation

Divergence

Curl

2d curl intuition

2d curl formula

2d curl example

2d curl nuance

Describing rotation in 3d with a vector

3d curl intuition, part 1

3d curl intuition, part 2

3d curl formula, part 1

Laplacian

Laplacian intuition

Laplacian computation example

Explicit Laplacian formula

Harmonic Functions

The Jacobian Determinant

Jacobian

Jacobian prerequisite knowledge

Local linearity for a multivariable function

The Jacobian matrix

Computing a Jacobian matrix

Tangent planes and local linearization

What is a tangent plane

Controlling a plane in space

Computing a tangent plane

Local linearization
Applications of multivariable derivatives

Quadratic approximations

What do quadratic approximations look like

Quadratic approximation formula, part 1

Quadratic approximation formula, part 2

Quadratic approximation example

The Hessian matrix

Expressing a quadratic form with a matrix

Vector form of multivariable quadratic approximation

Optimizing multivariable functions

Multivariable maxima and minima

Saddle points

Warm up to the second partial derivative test

Second partial derivative test

Second partial derivative test intuition

Second partial derivative test example, part 1

Second partial derivative test example, part 2

Lagrange multipliers and constrained optimization

Constrained optimization introduction

Lagrange multipliers, using tangency to solve constrained optimization

Finishing the intro lagrange multiplier example

Lagrange multiplier example, part 1

Lagrange multiplier example, part 2

The Lagrangian

Meaning of the Lagrange multiplier

Proof for the meaning of Lagrange multipliers

Line integrals for scalar functions (videos)

Introduction to the line integral

Line integral example 1

Line integral example 2 (part 1)

Line integral example 2 (part 2)

Line integrals in vector fields (videos)

Line integrals and vector fields

Using a line integral to find work

Parametrization of a reverse path

Scalar field line integral independent of path direction

Vector field line integrals dependent on path direction

Path independence for line integrals

Closed curve line integrals of conservative vector fields

Example of closed line integral of conservative field

Second example of line integral of conservative vector field
Integrating multivariable functions

Double integrals (videos)

Double integral 1

Double integral 2

Double integral 3

Double integral 4

Double integral 5

Double integral 6

Triple integrals (videos)

Triple integrals 1

Triple integrals 2

Triple integrals 3

Surface integral preliminaries (videos)

Parametrizing a surface, part 1

Determining a position vectorvalued function for a parametrization of two parameters

Partial derivatives of vectorvalued functions

Surface integrals (videos)

Introduction to the surface integral

Example of calculating a surface integral part 1

Example of calculating a surface integral part 2

Example of calculating a surface integral part 3

Surface integral example, part 1

Surface integral example part 2

Surface integral example part 3: The home stretch

Surface integral ex2 part 1

Surface integral ex2 part 2

Surface integral ex3 part 1

Surface integral ex3 part 2

Surface integral ex3 part 3

Surface integral ex3 part 4

Flux in 3D (videos)

Conceptual understanding of flux

Constructing a unit normal vector

Vector representation of a surface integral

Green’s theorem (videos)

Green’s theorem proof (part 1)

Green’s theorem proof (part 2)

Green’s theorem example 1

Green’s theorem example 2

2D divergence theorem

Constructing a unit normal vector to a curve

2D divergence theorem

Conceptual clarification for 2D divergence theorem

Stokes’ theorem (videos)

Stokes’ theorem intuition

Green’s and Stokes’ theorem relationship

Orienting boundary with surface

Orientation and stokes

Conditions for stokes theorem

Stokes example part 1

Stokes example part 2

Stokes example part 3

Stokes example part 4

Evaluating line integral directly – part 1

Evaluating line integral directly – part 2
Green’s, Stokes’, and the divergence theorems

3D divergence theorem (videos)

3D divergence theorem intuition

Divergence theorem example 1

Explanation of example 1

Proof of Stokes’ theorem

Stokes’ theorem proof part 1

Stokes’ theorem proof part 2

Stokes’ theorem proof part 3

Stokes’ theorem proof part 4

Stokes’ theorem proof part 5

Stokes’ theorem proof part 6

Stokes’ theorem proof part 7

Types of regions in three dimensions

Type I regions in three dimensions

Type II regions in three dimensions

Type III regions in three dimensions

Divergence theorem proof

Divergence theorem proof (part 1)

Divergence theorem proof (part 2)

Divergence theorem proof (part 3)

Divergence theorem proof (part 4)

Divergence theorem proof (part 5)
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