1. Geometry in higher dimensions
1a. Preliminaries
- The cross product
- The component formula for the cross product
- The scalar triple product
- Cross product examples
- Scalar triple product example
- Spherical coordinates
- Examples of n-dimensional vectors
- Function notation
1b. Lines and planes
- Parametrization of a line
- Forming planes
- A line or a plane or a point?
- Parametrization of a plane
- Distance from point to plane
- Parametrization of a line examples
- Forming plane examples
- Plane parametrization example
- Distance from point to plane example
- Intersecting planes example
1c. Matrices
- Matrices and determinants
- Multiplying matrices and vectors
- Dot product in matrix notation
- Matrices and linear functions
- The relationship between determinants and area or volume
- Matrix and vector multiplication examples
1d. Visualizing functions in higher dimensions
- Level sets
- Translation, rescaling, and reflection
- Surfaces of revolution
- Quadric surfaces
- Level set examples
2. Differential Calculus
2a. The partial derivative
- Introduction to partial derivatives
- Partial derivatives by limit definition
- Partial derivative examples
2b. The derivative and differentiability
- Differentiability in higher dimensions
- More on differentiability
- The linear approximation
- Differentiability examples
2c. The chain rule, directional derivative and gradient
- The chain rule
- Chain rule examples
- The directional derivative and the gradient
- Directional derivative and gradient examples
2d. Applications of differential calculus
3. Integral calculus
3a. Double integrals
- Double integrals
- Double integrals as iterated integrals
- Double integral examples
- Double integrals as volume or area
- Change order of integration examples
- Area of region example
- Changing order of integration is not always feasible
3b. Triple Integrals
3c. Changing variables
- Change of variables in double integrals
- Double integral change of variables examples
- Change of variables in triple integrals
- Triple integral change of variables examples
4. Vector calculus
4a. Vector field basics
- Vector fields
- The idea of divergence and curl
- The components of the curl
- More details on the components of the curl
- Divergence and curl notation
- Divergence and curl example
- Subtleties about divergence
- Subtleties about curl
4b. Parametrized curves
- Parametrized curves and their derivatives
- Tangent lines to parametrized curves
- Tangent line to parametrized curve examples
- Physical interpretation of a parametrization and its derivative
- The length of a path
- Length of a path examples
4c. Parametrized surfaces
- Parametrized surfaces
- Parametrized surface examples
- Surface area of parametrized surfaces
- Surface area calculation
- Normal vector of parametrized surfaces
- Orienting surfaces
- A surface that is not orientable
- Surface area example
4d. Path integrals and line integrals
- Path integral of a scalar-valued function
- Path integral examples
- Line integral of a vector field
- Alternate notation for line integrals
- Line integrals as circulation
- Line integral examples
4e. Surface integrals
5. The fundamental theorems of vector calculus
5a. Green's theorem
- The idea behind Green's theorem
- Other ways of writing Green's theorem
- Green's theorem with multiple boundary components
- Using Green's theorem to find area
- Sketch of proof for circulation per unit area
- Green's theorem examples
5b. The theorem for conservative vector fields
- Path-independent or conservative vector fields
- An example of a conservative vector field
- Path-independence implies no circulation
- Understanding the conditions for path-independence
- Finding the potential function
- Path-independence example in three dimensions