### Thread: Multivariable calculus

Covers basic pages in multivariable calculus

- Differential Calculus
- Partial derivatives
- 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
- The directional derivative and the gradient
- Applications of differential calculus

- Integral calculus
- Double integrals
- Triple Integrals
- 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
- Vector fields
- Vector field basics
- 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
- 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
- Stokes' theorem
- Divergence theorem
- Fundamental theorem synopsis

- Gradient theorem for line integrals