# UT Austin M 408M Multivariable Calculus: Study Guide, Practice

> A study guide for UT Austin M 408M, Multivariable Calculus: vectors, parametric and polar curves, partial derivatives, gradients and multiple integrals.

Source: https://practicecalculus.com/courses/ut-austin-m-408m
Updated: 2026-11-03

## At a glance

- **Course code:** M 408M (also offered as the 3-hour M 308M)
- **Hours:** 4 for M 408M (three lecture and two discussion hours a week); 3 for M 308M
- **Prerequisites:** M 408L or 408S with at least a C-
- **Covers:** Parametric equations, polar coordinates, vectors, vector calculus, partial derivatives, gradients and multiple integrals
- **Credit overlap:** Only one of M 403L, 408D and 408M (or 308M) may be counted
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- M 408M is UT Austin's course in the calculus of several variables, taken after M 408L or M 408S.
- Its catalog topics include parametric equations, polar coordinates, vectors, vector calculus, partial derivatives, gradients and multiple integrals.
- It is offered as the 4-hour M 408M or the 3-hour M 308M.
- M 408M and M 408D share a credit group, so only one of them may be counted.

**M 408M** is UT Austin's multivariable calculus course. It follows M 408L, Integral Calculus, and completes the three-course track that starts with M 408K. This guide maps every M 408M topic to free practice, so you can build intuition for vectors, partial derivatives and multiple integrals by solving problems with instant feedback.

## What UT Austin M 408M covers

According to the UT Austin catalog, M 408M introduces the theory and applications of differential and integral calculus of several variables. The listed topics include parametric equations, polar coordinates, vectors, vector calculus, partial derivatives, gradients and multiple integrals.

That is a standard **Calculus 3** course, with parametric curves and polar coordinates (often taught in Calculus 2 elsewhere) placed at the start. The 3-hour M 308M covers the same content without the discussion hours.

## M 408M topic map

| M 408M topic | Practice | Read |
| --- | --- | --- |
| Parametric equations | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |
| Polar coordinates | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Area in polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-area) |
| Vectors | [Vectors and their length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Curves and motion | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Chain rule](/rules/chain-rule) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Gradients | [The gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Optimization | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Multiple integrals | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Polar, cylindrical and spherical integrals | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |
| Line integrals and vector fields | [Work and line integrals](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Green's theorem](/theorems/greens-theorem) |
| Theorems of vector calculus | [Stokes and divergence](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [Divergence Theorem](/theorems/divergence-theorem) |

## M 408M vs M 408D

| | M 408M | M 408D |
| --- | --- | --- |
| Title | Multivariable Calculus | Sequences, Series, and Multivariable Calculus |
| Prerequisite | M 408L or 408S with a C- | M 408C, 408L or 408S with a C- |
| Main topics | Vectors, partial derivatives, gradients, multiple integrals, vector calculus | Integration methods, series, partial derivatives, multiple integrals |
| Hours | 4 (or 3 as M 308M) | 4 |

The two courses share a credit group. If you came through [M 408K](/courses/ut-austin-m-408k) and [M 408L](/courses/ut-austin-m-408l), M 408M is the natural next step; if you started with [M 408C](/courses/ut-austin-m-408c), see the [M 408D guide](/courses/ut-austin-m-408d).

## Example problems at the M 408M level

**1. A dot product angle.** For $\mathbf{u} = \langle 1, 2, 2 \rangle$ and $\mathbf{v} = \langle 2, 0, 0 \rangle$, $\mathbf{u} \cdot \mathbf{v} = 2$, $|\mathbf{u}| = 3$ and $|\mathbf{v}| = 2$, so

$$\cos\theta = \frac{2}{3 \cdot 2} = \frac{1}{3}$$

**2. A gradient.** For $f(x, y) = x^2 y + 3y$, the gradient is

$$\nabla f = \langle 2xy,\; x^2 + 3 \rangle, \qquad \nabla f(1, 2) = \langle 4, 4 \rangle$$

**3. A double integral in polar coordinates.** The area of the disk $r \le 2$ is

$$\int_0^{2\pi}\!\int_0^2 r\,dr\,d\theta = \int_0^{2\pi} 2\,d\theta = 4\pi$$

## How to do well in UT Austin M 408M

- **Get vectors solid first.** Nearly every later topic uses the dot and cross product. Start with the [vectors unit](/learn/calculus-3/vectors-and-space/vectors-and-length).
- **Sketch every region.** Most errors in multiple integrals are wrong limits, not wrong antiderivatives. Draw the region before you write the integral.
- **Remember the Jacobian factors.** Polar integrals need the extra $r$, and spherical integrals need $\rho^2 \sin\phi$. Our [formula sheet](/formulas) lists them.
- **Review the classic errors.** See [common calculus mistakes](/blog/common-calculus-mistakes) and the guide on [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam).
- **Practice a little each day** with [practice sets](/practice), mixing older topics in so they stay fresh.

## Frequently asked questions

### What is M 408M at UT Austin?

M 408M, Multivariable Calculus, introduces the theory and applications of differential and integral calculus of several variables. Its listed topics include parametric equations, polar coordinates, vectors, vector calculus, partial derivatives, gradients and multiple integrals.

### What do I need before M 408M?

The catalog prerequisite is M 408L or M 408S with a grade of at least C-. Students on the three-course track take M 408K, then M 408L, then M 408M.

### Should I take M 408M or M 408D?

M 408D is the second semester of the accelerated sequence and combines series with an introduction to several variables. M 408M is a full course on several variables, including vectors and vector calculus. Only one of M 403L, M 408D and M 408M may be counted, so the choice usually follows which first-year track you took.

### Is M 408M hard?

The new challenge is thinking in three dimensions: sketching surfaces, setting up limits of integration and keeping vectors straight. Students who draw a picture for every problem and practice steadily usually adjust well.

### How should I prepare for M 408M?

Review derivative rules and integration from M 408K and M 408L, then get comfortable with vectors, the dot product and the cross product before the course reaches partial derivatives.