# IU MATH-M 311 Calculus 3: Syllabus, Topics and Practice

> A study guide for Indiana University Bloomington MATH-M 311 Calculus 3: geometry of space, partial derivatives, max and min problems, multiple integrals.

Source: https://practicecalculus.com/courses/indiana-university-math-m311
Updated: 2026-11-08

## At a glance

- **Course code:** MATH-M 311
- **Credits:** 3 to 5 (as listed by IU)
- **Prerequisites:** Not listed on the IU course page; it follows MATH-M 212 Calculus II
- **Covers:** Elementary geometry of 2-, 3- and n-space, functions of several variables, partial differentiation, minimum and maximum problems, multiple integration
- **Offered by:** College of Arts and Sciences, IU Bloomington
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH-M 311 Calculus 3 is the multivariable calculus course at Indiana University Bloomington.
- It covers the geometry of 2-, 3- and n-space, functions of several variables, partial differentiation, max and min problems, and multiple integration.
- IU lists it at 3 to 5 credits.
- It follows MATH-M 211 and MATH-M 212.

**IU MATH-M 311** (Calculus 3) is the multivariable calculus course at Indiana University Bloomington and the third course after MATH-M 211 and MATH-M 212. IU lists it at 3 to 5 credits. This guide maps each topic in the official description to lessons you can practice with instant feedback.

## What IU MATH-M 311 covers

IU's official course listing describes MATH-M 311 as: elementary geometry of 2, 3, and n-space, functions of several variables, partial differentiation, minimum and maximum problems, multiple integration.

That covers the core of **Calculus 3**: vectors and the geometry of lines and planes, functions of two or more variables, partial derivatives and gradients, optimization including constrained problems, and double and triple integrals. The listing does not mention line integrals, surface integrals or Green's and Stokes' theorems, so treat those as extras unless your syllabus includes them. The listing also does not give a textbook or schedule.

## MATH-M 311 topic map

| M311 topic | Practice | Read |
| --- | --- | --- |
| Points and distance in space | [Points and distance](/learn/calculus-3/vectors-and-space/points-and-distance) | [Vectors and length](/learn/calculus-3/vectors-and-space/vectors-and-length) |
| Vectors and dot product | [Dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Cross product | [Cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Projections and work](/learn/calculus-3/vectors-and-space/projections-and-work) |
| Lines and planes | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial differentiation | [Partials with rules](/learn/calculus-3/partial-derivatives/partials-with-rules) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Chain rule and gradient | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Max and min problems | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Constrained max and min | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Changing the order and polar form | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) |

## M311 in the IU Bloomington sequence

| Course | Credits | Covers |
| --- | --- | --- |
| [MATH-M 211 Calculus I](/courses/indiana-university-math-m211) | 4 | Limits, derivatives, integrals, applications |
| [MATH-M 212 Calculus II](/courses/indiana-university-math-m212) | 4 | Integration techniques, improper integrals, applications, series |
| MATH-M 311 Calculus 3 | 3 to 5 | Several variables, partial derivatives, multiple integrals |

## How M311 builds on M211 and M212

Nearly every M311 skill reuses one-variable calculus. A partial derivative is an ordinary derivative with the other variables held fixed, and a double integral is two single integrals done one after the other. That makes it worth reviewing derivative rules and integration techniques before the term starts.

## Example problems at the M311 level

**1. Distance in 3-space.** Between $(1, 2, 3)$ and $(4, 6, 3)$:

$$\sqrt{(4-1)^2 + (6-2)^2 + 0^2} = 5$$

**2. A constrained maximum.** Maximize $f = xy$ subject to $x + y = 10$. Lagrange multipliers give $y = \lambda$ and $x = \lambda$, so $x = y = 5$ and the maximum is $25$.

**3. A double integral over a triangle.**

$$\int_0^1 \int_0^x 2\,dy\,dx = \int_0^1 2x\,dx = 1$$

## How to do well in MATH-M 311

- **Think geometrically.** Picture vectors as arrows and functions of two variables as surfaces.
- **Treat other variables as constants** when taking a partial derivative. Practice with [partial derivatives practice](/learn/calculus-3/partial-derivatives/partial-derivatives-practice).
- **Draw the region first** for every double or triple integral, then choose the order of integration.
- **Practice [Lagrange multipliers problems](/practice-problems/lagrange-multipliers)** and [double integral problems](/practice-problems/double-integrals).
- **Use timed [practice exams](/practice)** and the IU Math Learning Center before tests.

> **Tip:** at a critical point of $f(x, y)$, use $D = f_{xx}f_{yy} - f_{xy}^2$. If $D > 0$ and $f_{xx} > 0$ it is a minimum, if $D > 0$ and $f_{xx} < 0$ a maximum, and if $D < 0$ a saddle.

## Frequently asked questions

### What is MATH-M 311 at Indiana University?

MATH-M 311 Calculus 3 is IU Bloomington's multivariable calculus course. IU lists elementary geometry of 2, 3 and n-space, functions of several variables, partial differentiation, minimum and maximum problems, and multiple integration.

### How many credits is MATH-M 311?

IU lists MATH-M 311 at 3 to 5 credits. Check the schedule of classes for the credit value of your section.

### What do I need before MATH-M 311?

The IU course page does not print a prerequisite. M311 follows MATH-M 212 Calculus II, so plan on finishing M212 first and confirm with your advisor.

### Does M311 cover Green's and Stokes' theorems?

The official description does not mention line integrals, surface integrals or the theorems of vector calculus. It ends with multiple integration. Your section syllabus will say whether any vector calculus is added.

### Why does M311 mention n-space?

The course treats vectors and geometry in any number of dimensions, not just 2 and 3. The ideas, like dot products and distance, work the same way in higher dimensions.