Columbia MATH UN1202 Calculus IV: Topics, Study Guide and Practice

Columbia MATH UN1202 is Calculus IV at Columbia University, the second half of the department's multivariable sequence. Students take it after both Calculus II and Calculus III, and it covers the integration side of multivariable calculus, vector calculus and an introduction to Fourier series. This guide maps every UN1202 topic to practice with instant feedback.

What Columbia MATH UN1202 covers

According to the Columbia bulletin, UN1202 covers multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, and Fourier series.

That is the second half of a standard Calculus 3 (multiple integrals and vector calculus, through Green's, Stokes' and the Divergence Theorem), plus Fourier series, which many universities leave for a differential equations course.

UN1202 topic map

We do not have dedicated Fourier series lessons yet. Computing Fourier coefficients is mostly integration by parts with sines and cosines, so that lesson is the best preparation.

The standard Columbia sequence

Students who want to move faster can take UN1205 Accelerated Multivariable Calculus instead of UN1201 and UN1202. Credit is not given for both.

Example problems at the UN1202 level

1. A double integral over a triangle. Over the region 0 \le y \le x \le 1:

2. A conservative vector field. \mathbf{F} = \langle 2xy, x^2 \rangle = \nabla(x^2 y), so for any path from (0, 0) to (2, 3),

3. A Fourier sine coefficient. For f(x) = x on (-\pi, \pi), the function is odd, so only sine terms appear. Integration by parts gives

so x = 2\left(\sin x - \frac{\sin 2x}{2} + \frac{\sin 3x}{3} - \cdots\right) on (-\pi, \pi).

How to do well in MATH UN1202

• Set up before you integrate. Draw every region and decide the order of integration from the picture. The general regions lesson builds the habit.
• Remember the Jacobian factors: r for polar and cylindrical, \rho^2 \sin\phi for spherical.
• Learn the vector calculus theorems together. Green's theorem, Stokes' theorem and the Divergence Theorem are one idea in three settings.
• Keep integration by parts sharp for Fourier series, using the formula sheet, and rehearse with timed practice exams.

Tip: before computing a Fourier coefficient, check whether the function is even or odd. Half of the coefficients are zero for free.