Multivariable calculus | 1 Big School (By Blue Brain)

Course Content

Thinking about multivariable functions

Introduction to multivariable calculus
Multivariable functions
Visualizing scalar-valued functions
Representing points in 3d
Introduction to 3d graphs
Interpreting graphs with slices
Contour plots
Visualizing vector-valued 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
Vector-valued functions intro
Vector-valued 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 vector-valued 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 vector-valued 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 vector-valued function for a parametrization of two parameters
Partial derivatives of vector-valued 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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Course Content

Thinking about multivariable functions

Introduction to multivariable calculus

Multivariable functions

Visualizing scalar-valued functions

Representing points in 3d

Introduction to 3d graphs

Interpreting graphs with slices

Contour plots

Visualizing vector-valued 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

Vector-valued functions intro

Vector-valued 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 vector-valued 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 vector-valued 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