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

> A study guide for Columbia MATH UN1202 Calculus IV: multiple integrals, line and surface integrals, vector field calculus and Fourier series, with practice.

Source: https://practicecalculus.com/courses/columbia-math-un1202
Updated: 2026-11-03

## At a glance

- **Course code:** MATH UN1202
- **Points:** 3
- **Offered:** Fall and spring
- **Prerequisites:** MATH UN1102 and MATH UN1201, or the equivalent
- **Covers:** Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series
- **Then:** MATH UN2010 Linear Algebra completes the standard sequence
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH UN1202 is Columbia's Calculus IV, the second half of its multivariable sequence, worth 3 points.
- It requires both Calculus II (UN1102) and Calculus III (UN1201).
- It covers multiple integrals, Taylor's formula in several variables, line and surface integrals, vector field calculus and Fourier series.
- UN1205 Accelerated Multivariable Calculus combines UN1201 and UN1202 into one course.

**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

| UN1202 topic | Practice | Read |
| --- | --- | --- |
| Double integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| General regions and order of integration | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) | [Double integrals](/blog/double-integrals) |
| Polar, cylindrical and spherical | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |
| Change of variables | [Change of variables and the Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [Double integrals](/blog/double-integrals) |
| Taylor's formula in several variables | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Taylor's theorem](/theorems/taylors-theorem), [Taylor series](/blog/taylor-series) |
| Vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Green's theorem](/theorems/greens-theorem) |
| Curl and divergence | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Divergence Theorem](/theorems/divergence-theorem) |
| Surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Stokes' theorem](/theorems/stokes-theorem) |
| Fourier series | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts), [sine function](/functions/sine-function) |

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

| Course | Role |
| --- | --- |
| [UN1101 Calculus I](/courses/columbia-math-un1101) | Limits, derivatives, intro to integrals |
| [UN1102 Calculus II](/courses/columbia-math-un1102) | Integration methods, Taylor's theorem, series |
| [UN1201 Calculus III](/courses/columbia-math-un1201) | Vectors, partial derivatives, optimization |
| UN1202 Calculus IV | Multiple integrals, vector calculus, Fourier series |
| UN2010 Linear Algebra | Completes the 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 $0 \le y \le x \le 1$:

$$\int_0^1 \int_0^x 2y\,dy\,dx = \int_0^1 x^2\,dx = \frac{1}{3}$$

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

$$\int_C \mathbf{F} \cdot d\mathbf{r} = (2)^2(3) - 0 = 12$$

**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

$$b_n = \frac{2}{\pi} \int_0^{\pi} x \sin(nx)\,dx = \frac{2(-1)^{n+1}}{n}$$

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](/learn/calculus-3/multiple-integrals/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](/theorems/greens-theorem), [Stokes' theorem](/theorems/stokes-theorem) and the [Divergence Theorem](/theorems/divergence-theorem) are one idea in three settings.
- **Keep integration by parts sharp** for Fourier series, using the [formula sheet](/formulas), and rehearse with timed [practice exams](/practice).

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

## Frequently asked questions

### What is Columbia MATH UN1202?

MATH UN1202 is Calculus IV at Columbia. It covers multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields and Fourier series.

### Why does Columbia have a Calculus IV?

Columbia splits multivariable calculus into two 3-point courses. UN1201 covers vectors and differential calculus in several variables, and UN1202 covers integration in several variables and vector calculus, plus Fourier series.

### What do I need before UN1202?

The bulletin requires both MATH UN1102 Calculus II and MATH UN1201 Calculus III, or the equivalent.

### What comes after UN1202?

In the department's standard sequence, UN1202 is followed by MATH UN2010 Linear Algebra. Check your major for further requirements such as differential equations.

### Is Calculus IV harder than Calculus III?

It is more computational, with lots of integral setups in different coordinate systems, and the big theorems of vector calculus. Students who sketch regions carefully and keep their single-variable integration sharp usually find it manageable.