We talk about basic geometry in higher dimensions.

Learn how to draw a sphere.

Vectors are lists of numbers that denote direction and magnitude.

The dot product measures how aligned two vectors are with each other.

The cross product is a special way to multiply two vectors in three-dimensional space.

Vector-valued functions are parameterized curves.

With one input, and vector outputs, we work component-wise.

We interpret vector-valued functions as paths of objects in space.

We find a new description of curves that trivializes arc length computations.

We introduce two important unit vectors.

We discuss how to find implicit and explicit formulas for planes.

Tangent and normal vectors can help us make interesting parametric plots.

Learn how to draw a torus.

We introduce functions that take vectors or points as inputs and output a number.

We introduce level sets.

We investigate what continuity means for real-valued functions of several variables.

We introduce partial derivatives and the gradient vector.

We use the gradient to approximate values for functions of several variables.

We find tangent planes.

We introduce differentiability for functions of several variables and find tangent planes.

We introduce a way of analyzing the rate of change in a given direction.

We investigate the chain rule for functions of several variables.

The gradient is the fundamental notion of a derivative for a function of several variables.

We introduce Taylor polynomials for functions of several variables.

We will get to know some basic quadric surfaces.

Learn how to draw an elliptic and a hyperbolic paraboloid.

We see how to find extrema of functions of several variables.

We learn to optimize surfaces along and within given paths.

We give a new method of finding extrema.

We study integrals over basic regions.

We study integrals over general regions by integrating $1$.

We integrate over regions in polar coordinates.

We integrate over regions in cylindrical coordinates.

We integrate over regions in spherical coordinates.

We compute surface area with double integrals.

We use integrals to model mass.

We practice more computations and think about what integrals mean.

We introduce the idea of a vector at every point in space.

We accumulate vectors along a path.

Green’s Theorem is a fundamental theorem of calculus.

A planimeter computes the area of a region by tracing the boundary.

Divergence measures the rate field vectors are expanding at a point.

We generalize the idea of line integrals to higher dimensions.

Learn how to draw a Möbius strip.

We introduce the divergence theorem.

We introduce Stokes’ theorem.

We explore the relationship between the gradient, the curl, and the divergence of a vector field.