Math 223 – Calculus III
Math 223 is a one semester course in Multivariable Calculus, addressing vectors and multivariable
functions, partial derivatives and gradients, double and triple integrals in various coordinate systems,
vector fields, line integrals, flux, divergence and Stokes' Theorem. Students who successfully complete
this course will be expected to:

Use vectors as a tool for working with ℝ𝑛 , where 𝑛 ≥ 3. In particular, students are expected to compute or
apply vector addition, scaling, dot and cross products within the context of broader questions.

Identify properties of a function 𝑓(𝑥, 𝑦), (zeros, limiting behavior, periodicity, etc.) that allow students to
visualize the graph 𝑧 = 𝑓(𝑥, 𝑦). This also includes sketching and/or interpreting traces and level sets.

Compute partial derivatives and directional derivatives of elementary functions, and interpret the graphical
meaning of partial derivatives and directional derivatives at a point.

Find and use the gradient of a function as applicable to tangent planes, normal lines, and maximal
directional derivatives.

Approximate volumes bounded by a rectangle in the xy-plane and a surface 𝑧 = 𝑓(𝑥, 𝑦) using Riemann
sums, and determine whether ∫ ∫ 𝑅.𝑓(𝑥, 𝑦) 𝑑𝐴 is positive, negative, or zero without computing the integral.

Set up and evaluate double integrals over regions in the xy-plane bounded by curves 𝑦 = 𝑓(𝑥), switch the
order of integration for double integrals, and convert between rectangular and polar coordinates for double
integrals.

Set up, evaluate, and change coordinates for triple integrals in rectangular, cylindrical, and spherical
coordinates.

Identify how properties of a function 𝐹⃗ (𝑥, 𝑦) affect the graph of a vector field; parameterize line segments,
graphs of functions, and elliptical arcs within ℝ2 and ℝ3 and use parameterizations to compute line
integrals.

Apply the Fundamental Theorem of Line Integrals and Green's Theorem to compute line integrals, and
identify the appropriate times to use these theorems.

Compute the flux of a constant vector field through a flat surface without integration, and use integration to
compute flux of non-constant vector fields through surfaces 𝑧 = 𝑓(𝑥, 𝑦), cylinders, or spheres, and when
applicable use the divergence theorem to compute flux.

Find the curl of a vector field, and use Stokes' Theorem to find the circulation of a vector field around a
simple closed curve in ℝ3 .