Multivariable Calculus

Functions of Several Variables

Domain and Range

Level Curves and Contour Maps

Partial Differentiation

Limits and Continuity in ${\mathbb{R}}^n$

Partial Derivatives definition

Total Differential

Linear Approximation (Taylor series)

The Chain Rule for several variables

Implicit Differentiation

Extrema of Multivariable Functions

Critical Points

Hessian Matrix and Determinant test

Saddle Points, Local Max/Min

Lagrange Multipliers (Constrained Optimization)

Parametric Curves

Cartesian to Vector-valued functions $\vec{r}(t)$

Parametrization of various curves (circle, ellipse, helicoid)

Velocity, Acceleration, and Tangent Vectors

Arc and curve Length

Archimedes spiral

Curvature

Frenet Frame (Tangent, Normal, Binormal) ?

Vector Calculus

Line integral (work)

Conservative Fields and Potential Functions

The Gradient Vector $\nabla$

Directional Derivatives

Divergence and Curl

Multiple Integrals

Double Integrals

Rectangular Coordinates

Polar Coordinates

Change of variables

Applications

Triple Integrals

Cartesian Coordinates

Cylindrical Coordinates

Spherical Coordinates

Change of variables

Applications

The Jacobian Determinant

Change of Variables

Handwritten