A sequence is an ordered list of numbers.

A sequence can be thought of as a function from the integers to the real numbers. There are two ways to establish whether a sequence has a limit.

A series is summation of a sequence.

Infinite sums can be studied using improper integrals.

If an infinite sum converges, then its terms must tend to zero.

Some infinite series can be compared to geometric series.

Some infinite series can be compared to geometric series.

We compare infinite series to each other using inequalities.

We compare infinite series to each other using limits.

Alternating series have nice properties.

We can approximate smooth functions with polynomials.

Infinite series can represent functions.

We study Taylor and Maclaurin series.

Taylor series are a computational tool.

Power series interact nicely with other calculus concepts.

We discuss the basics of parametric curves.

We discuss derivatives and integrals of parametric curves.

Polar coordinates are a special type of parametric curves.

We see a collection of polar curves.

We differentiate polar functions.

We integrate polar functions.

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 plot make interesting parametric functions.

Learn how to draw a torus.

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

We investigate what continuity means for functions of several variables.

We introduce partial derivatives.

We introduce the gradient vector.

We extend the ideas of linear approximation to functions of several variables.

We investigate the chain rule for functions 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 with and integrand equaling one.

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.

We introduce the divergence theorem.

We introduce Stokes’ theorem.