# NYU MATH-UA 123 Calculus III: Topics, Study Guide and Practice

> A study guide for NYU MATH-UA 123 Calculus III: vectors, partial derivatives, multiple integrals, line and surface integrals and Stokes, with practice.

Source: https://practicecalculus.com/courses/nyu-math-ua-123
Updated: 2026-11-03

## At a glance

- **Course code:** MATH-UA 123
- **Credits:** 4
- **Offered:** Fall, spring and summer
- **Prerequisites:** A grade of C or better in MATH-UA 122, a 5 on AP Calculus BC, or a qualifying placement score
- **Covers:** Functions of several variables, vectors, partial derivatives, double and triple integrals, line and surface integrals, the theorems of Gauss and Stokes
- **Honors version:** MATH-UA 129 Honors Calculus III
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH-UA 123 is NYU's Calculus III, a 4-credit multivariable calculus course offered in fall, spring and summer.
- It covers vectors, partial derivatives, double and triple integrals, line and surface integrals, and the theorems of Gauss and Stokes.
- A 5 on AP Calculus BC can place you directly into MATH-UA 123.
- MATH-UA 129 is an honors version that covers the same core at a faster pace and in more depth.

**NYU MATH-UA 123** is Calculus III at NYU, the multivariable course that completes the College of Arts and Science calculus sequence. Math, physics, computer science and economics students take it after Calculus II, and it is the gateway to courses like linear algebra applications, differential equations and physics. This guide maps each MATH-UA 123 topic to Calculus 3 practice with instant feedback.

## What NYU MATH-UA 123 covers

The NYU bulletin lists functions of several variables, vectors, partial derivatives, double and triple integrals, and line and surface integrals, finishing with the theorems of Gauss (the Divergence Theorem) and Stokes.

That is a complete **Calculus 3**: vectors and space, differential calculus in several variables, multiple integration, and vector calculus. It lines up with all four units of the Calculus 3 path on PracticeCalculus.

## MATH-UA 123 topic map

| MATH-UA 123 topic | Practice | Read |
| --- | --- | --- |
| Vectors, dot and cross products | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines, planes and curves | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| 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) |
| Gradient and directional derivatives | [The gradient](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [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) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integrals](/blog/double-integrals) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals, Gauss and Stokes | [Stokes and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [Divergence Theorem](/theorems/divergence-theorem) |

## MATH-UA 123 vs MATH-UA 129

| Course | Pace and depth | Entry (from the bulletin) |
| --- | --- | --- |
| MATH-UA 123 Calculus III | Standard multivariable course | C in MATH-UA 122, AP BC 5, or placement |
| MATH-UA 129 Honors Calculus III | Same core, faster and deeper, includes Lagrange multipliers | A- in MATH-UA 122, AP BC 5, or placement |

The two courses exclude each other, so pick one. Before either, the earlier courses in the sequence are [MATH-UA 121 Calculus I](/courses/nyu-math-ua-121) and [MATH-UA 122 Calculus II](/courses/nyu-math-ua-122).

## Example problems at the MATH-UA 123 level

**1. Partial derivatives.** For $f(x, y) = x^2 y + 3y^3$:

$$f_x = 2xy, \qquad f_y = x^2 + 9y^2$$

**2. An iterated integral.**

$$\int_0^1 \int_0^2 xy\,dy\,dx = \int_0^1 x \left[\frac{y^2}{2}\right]_0^2 dx = \int_0^1 2x\,dx = 1$$

**3. A directional derivative.** For $f(x, y) = x^2 + y^2$ at $(1, 2)$ in the direction of $\langle 3, 4 \rangle$: the gradient is $\nabla f = \langle 2x, 2y \rangle = \langle 2, 4 \rangle$, and the unit vector is $\langle \frac{3}{5}, \frac{4}{5} \rangle$, so

$$D_{\mathbf{u}} f = 2 \cdot \frac{3}{5} + 4 \cdot \frac{4}{5} = \frac{22}{5}$$

## How to do well in MATH-UA 123

- **Sketch everything.** Most mistakes in multiple integrals are wrong bounds. Draw the region before you write a single limit of integration, and practice [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order).
- **Know when to change coordinates.** Circles and spheres call for [polar](/learn/calculus-3/multiple-integrals/polar-double-integrals) or [cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) coordinates. Do not forget the extra $r$ or $\rho^2 \sin\phi$.
- **Connect the big theorems.** [Green's theorem](/theorems/greens-theorem), [Stokes' theorem](/theorems/stokes-theorem) and the [Divergence Theorem](/theorems/divergence-theorem) are one idea in three settings. Learn them side by side.
- **Keep your single-variable skills sharp** with the [formula sheet](/formulas), and test yourself with timed [practice exams](/practice).

> **Tip:** before computing a line integral, check whether the field is conservative. If it is, the answer is just the potential function evaluated at the endpoints.

## Frequently asked questions

### What is NYU MATH-UA 123?

MATH-UA 123 is Calculus III at NYU, the multivariable course. It covers functions of several variables, vectors, partial derivatives, double and triple integrals, and line and surface integrals, ending with the theorems of Gauss and Stokes.

### What is the difference between MATH-UA 123 and MATH-UA 129?

MATH-UA 129 Honors Calculus III covers the same core material faster and in more depth, including Lagrange multipliers. The bulletin asks for an A- or better in MATH-UA 122, a 5 on AP Calculus BC, or a placement score, and it is offered only occasionally. Students take one or the other, not both.

### Do I need MATH-UA 122 before MATH-UA 123?

Usually yes: the bulletin lists a C or better in MATH-UA 122. Students with a 5 on AP Calculus BC or a qualifying placement score can enter directly.

### Is Calculus III harder than Calculus II?

Calculus III asks you to think in three dimensions, but most of its computations reuse single-variable skills: a partial derivative is an ordinary derivative, and a double integral is two ordinary integrals. Students who are comfortable with integration usually adapt quickly.

### How should I prepare for MATH-UA 123?

Refresh integration techniques and polar coordinates from Calculus II, then get comfortable with vectors, the dot product and the cross product. Those tools appear in almost every later topic.