# Columbia MATH UN1201 Calculus III: Topics, Study Guide and Practice

> A study guide for Columbia MATH UN1201 Calculus III: vectors, partial derivatives, gradients, optimization and Lagrange multipliers, with practice.

Source: https://practicecalculus.com/courses/columbia-math-un1201
Updated: 2026-11-03

## At a glance

- **Course code:** MATH UN1201
- **Points:** 3
- **Offered:** Fall and spring
- **Prerequisites:** MATH UN1101 or the equivalent (UN1102 strongly urged first)
- **Covers:** Vectors in two and three dimensions, complex numbers and the complex exponential, Cramer's rule, vector-valued functions, functions of several variables, partial derivatives, gradients, surfaces, optimization, Lagrange multipliers
- **Next course:** MATH UN1202 Calculus IV
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH UN1201 is Columbia's Calculus III, the first half of its two-semester multivariable sequence, worth 3 points.
- It covers vectors, complex numbers and the complex exponential, vector-valued functions, partial derivatives, gradients, optimization and Lagrange multipliers.
- Multiple integrals and vector calculus come next, in UN1202 Calculus IV.
- A 5 on AP Calculus BC lets students begin with Calculus III.

**Columbia MATH UN1201** is Calculus III at Columbia University, the first half of the department's two-course multivariable sequence. Students take it after Calculus I and usually Calculus II, and it is unusual in covering complex numbers and Cramer's rule alongside vectors and partial derivatives. This guide maps every UN1201 topic to practice with instant feedback.

## What Columbia MATH UN1201 covers

The Columbia bulletin lists vectors in two and three dimensions, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, and the method of Lagrange multipliers.

That is the first half of a standard **Calculus 3**: vectors, curves in space and differential calculus in several variables. Multiple integrals and vector calculus wait for UN1202. The complex exponential and Cramer's rule are extras you will not find in most Calculus 3 courses.

## UN1201 topic map

| UN1201 topic | Practice | Read |
| --- | --- | --- |
| Vectors in 2D and 3D | [Vectors and their length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Dot and cross products | [The cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Complex exponential | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [natural exponential function](/functions/natural-exponential-function) |
| Vector-valued functions | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Partial derivatives](/blog/partial-derivatives) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Gradients and surfaces | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Optimization | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization problems](/blog/optimization-problems) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |

## UN1201, UN1202 and UN1205

| Course | Points | Covers |
| --- | --- | --- |
| UN1201 Calculus III | 3 | Vectors, complex exponential, partial derivatives, optimization |
| [UN1202 Calculus IV](/courses/columbia-math-un1202) | 3 | Multiple integrals, line and surface integrals, vector fields, Fourier series |
| UN1205 Accelerated Multivariable Calculus | 4 | The UN1201 and UN1202 material in one semester |

UN1202 requires both [UN1102 Calculus II](/courses/columbia-math-un1102) and UN1201, so if you started at UN1201 without Calculus II, plan for it before Calculus IV.

## Example problems at the UN1201 level

**1. A cross product.**

$$\langle 1, 2, 0 \rangle \times \langle 0, 1, 3 \rangle = \langle 2 \cdot 3 - 0 \cdot 1,\ 0 \cdot 0 - 1 \cdot 3,\ 1 \cdot 1 - 2 \cdot 0 \rangle = \langle 6, -3, 1 \rangle$$

**2. The complex exponential and a differential equation.** For $y'' + y = 0$, try $y = e^{rx}$: then $r^2 + 1 = 0$, so $r = \pm i$. Using $e^{ix} = \cos x + i\sin x$, the real solutions are

$$y = c_1 \cos x + c_2 \sin x$$

**3. Lagrange multipliers.** Maximize $f = xy$ subject to $x + y = 10$. From $\nabla f = \lambda \nabla g$: $y = \lambda$ and $x = \lambda$, so $x = y = 5$ and the maximum is $f = 25$.

## How to do well in MATH UN1201

- **Get vectors right first.** Every later topic uses the [dot product](/learn/calculus-3/vectors-and-space/dot-product) and cross product, so make them automatic in the first weeks.
- **Revisit Taylor series for the complex exponential.** Euler's formula falls out of the series for $e^x$, $\sin x$ and $\cos x$. Our [Taylor series guide](/blog/taylor-series) covers them.
- **Treat partial derivatives as ordinary derivatives** with the other variables frozen. The usual [chain rule](/rules/chain-rule) and [product rule](/rules/product-rule) still apply.
- **Keep the [formula sheet](/formulas) handy** and use timed [practice exams](/practice) before each midterm.

> **Tip:** in a Lagrange multiplier problem, always check the boundary or the ends of the constraint too. The method finds candidates, and you still compare values.

## Frequently asked questions

### What is Columbia MATH UN1201?

MATH UN1201 is Calculus III at Columbia. It covers vectors in two and three dimensions, complex numbers and the complex exponential with applications to differential equations, Cramer's rule, vector-valued functions, functions of several variables, partial derivatives, gradients, surfaces, optimization and Lagrange multipliers.

### Does Columbia Calculus III include multiple integrals?

No. At Columbia the multivariable material is split over two courses. UN1201 covers differential calculus in several variables, and UN1202 Calculus IV covers multiple integrals, line and surface integrals and vector field calculus.

### Can I take UN1201 without Calculus II?

The bulletin lists UN1101 or the equivalent as the prerequisite and says Calculus II is no longer required, but the department strongly urges taking it first, especially for prospective math majors. Students with a 5 on AP Calculus BC may begin with Calculus III.

### What is the difference between UN1201 and UN1205?

UN1205 Accelerated Multivariable Calculus covers the material of UN1201 and UN1202 in one 4-point course and requires UN1101 and UN1102. Credit is not given for both UN1205 and the UN1201 and UN1202 pair.

### Is UN1201 hard?

It introduces several new kinds of objects at once: vectors, complex numbers and functions of several variables. Each topic is approachable on its own, so steady weekly practice matters more than cramming.