UCSD MATH 20C Multivariable Calculus: Study Guide and Practice

MATH 20C is the third course in UC San Diego's calculus sequence for science and engineering, and the point where calculus moves from one variable to several. Most students reach it after MATH 20B; a strong AP Calculus BC score also qualifies. This guide maps every MATH 20C topic to free practice so you can build intuition for vectors, surfaces and double integrals one problem at a time.

What UCSD MATH 20C covers

The UC San Diego catalog lists five topics for MATH 20C: vector geometry, vector functions and their derivatives, partial differentiation, maxima and minima, and double integration. The course was formerly numbered MATH 21C.

That matches the first two thirds of a standard Calculus 3 course. Triple integrals, line integrals and the theorems of Green, Stokes and Gauss are not part of the 20C description. UC San Diego lists those vector calculus ideas in a separate course, MATH 20E (Vector Calculus).

MATH 20C topic map

Each row links to a lesson you can practice and to a reference page or guide that explains the idea.

MATH 20C and the courses around it

If you are coming from AP Calculus BC with a 4 or 5, you can start the sequence here. Otherwise, the MATH 20B guide and the MATH 20A guide cover the single-variable skills 20C relies on.

Example problems at the MATH 20C level

1. The angle between two vectors. Find the angle between \mathbf{u} = \langle 1, 1, 0 \rangle and \mathbf{v} = \langle 1, 0, 1 \rangle.

\mathbf{u} \cdot \mathbf{v} = 1 and |\mathbf{u}| = |\mathbf{v}| = \sqrt{2}, so

2. Partial derivatives. For f(x, y) = x^2 y + \sin(xy),

3. A double integral. Evaluate \iint_R xy\,dA over R = [0, 2] \times [0, 3].

How to do well in UCSD MATH 20C

• Sketch everything. Draw the region before setting up a double integral and the vectors before computing an angle. Most setup errors disappear once you can see the picture.
• Lean on single-variable rules. A partial derivative is an ordinary derivative with the other variables held constant, so the product rule and chain rule carry over directly.
• Practice the second-derivative test until classifying critical points is routine, then mix in constrained problems with Lagrange multipliers.
• Use the formula sheet and timed practice. The calculus formula sheet and practice exams help you get faster before midterms.

Tip: when a double integral looks messy in x and y, check whether the region is a disk or a wedge. Switching to polar coordinates often turns a hard integral into a one-line answer.