# UF MAC 2313 Calculus 3: Topics, Study Guide and Practice

> A study guide for UF MAC 2313 Analytic Geometry and Calculus 3: solid analytic geometry, vectors, partial derivatives and multiple integrals, with practice.

Source: https://practicecalculus.com/courses/university-of-florida-mac-2313
Updated: 2026-11-03

## At a glance

- **Course code:** MAC 2313
- **Credits:** 4
- **Prerequisites:** MAC 2312, MAC 2512 or MAC 3473 with a minimum grade of C
- **Covers:** Solid analytic geometry, vectors, partial derivatives, multiple integrals
- **Honors version:** MAC 3474 Honors Calculus 3
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MAC 2313 is Calculus 3 at the University of Florida, a 4-credit General Education mathematics course.
- The catalog lists solid analytic geometry, vectors, partial derivatives and multiple integrals.
- It requires a C or better in MAC 2312, MAC 2512 or MAC 3473.
- Credit is given for at most one of MAC 2313 and MAC 3474, the honors version.

**UF MAC 2313** (Analytic Geometry and Calculus 3) is the multivariable calculus course at the University of Florida and the last of the MAC 2311, 2312, 2313 sequence, covering the geometry of space and calculus in several variables. This guide maps each MAC 2313 topic to Calculus 3 practice with instant feedback.

## What UF MAC 2313 covers

The UF catalog description is four items long: solid analytic geometry, vectors, partial derivatives and multiple integrals. "Solid analytic geometry" means the geometry of three-dimensional space: distance, lines, planes and surfaces.

In standard terms that is the core of **Calculus 3**: vectors and space, partial derivatives, and double and triple integrals. On PracticeCalculus that is the first three units of the Calculus 3 path. If your section also covers vector fields, line integrals or Green's theorem, the vector calculus unit has those too.

## MAC 2313 topic map

| MAC 2313 topic | Practice | Read |
| --- | --- | --- |
| Points and distance in space | [Points and distance in space](/learn/calculus-3/vectors-and-space/points-and-distance) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vectors | [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 dot product](/learn/calculus-3/vectors-and-space/dot-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) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives guide](/blog/partial-derivatives) |
| Higher-order partials | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-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) |
| Extrema and Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Polar double integrals | [Double integrals in polar coordinates](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals](/blog/double-integrals) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integrals](/blog/double-integrals) |

## MAC 2313 in the UF sequence

| Course | Credits | Covers |
| --- | --- | --- |
| [MAC 2311 Calculus 1](/courses/university-of-florida-mac-2311) | 4 | Limits, derivatives, intro to integration |
| [MAC 2312 Calculus 2](/courses/university-of-florida-mac-2312) | 4 | Integration techniques and applications, series |
| MAC 2313 Calculus 3 | 4 | Solid analytic geometry, vectors, partials, multiple integrals |

The honors track runs MAC 3472, 3473 and 3474 in parallel, covering the same topics in greater depth. Credit is given for only one of MAC 2313 and MAC 3474.

## Example problems at the MAC 2313 level

**1. Distance in space.** The distance between $(1, 2, 3)$ and $(4, 6, 3)$ is

$$\sqrt{(4 - 1)^2 + (6 - 2)^2 + (3 - 3)^2} = \sqrt{9 + 16} = 5$$

**2. A mixed partial derivative.** For $f(x, y) = x^3 y^2$: $f_x = 3x^2 y^2$, so

$$f_{xy} = \frac{\partial}{\partial y}\left(3x^2 y^2\right) = 6x^2 y$$

Computing $f_y = 2x^3 y$ and then $f_{yx} = 6x^2 y$ gives the same answer, as Clairaut's theorem predicts.

**3. A double integral.**

$$\int_0^1 \int_0^2 (x + y)\,dy\,dx = \int_0^1 \left[xy + \frac{y^2}{2}\right]_0^2 dx = \int_0^1 (2x + 2)\,dx = 3$$

## How to do well in MAC 2313

- **Start with vectors and keep them sharp.** The [cross product](/learn/calculus-3/vectors-and-space/cross-product) gives normal vectors to planes, which you will need for tangent planes later.
- **Sketch every region of integration.** Most mistakes in multiple integrals are wrong bounds, not wrong antiderivatives. Practice [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order).
- **Use polar coordinates for circles.** And remember the extra $r$ in $dA = r\,dr\,d\theta$.
- **Keep single-variable skills current** with the [formula sheet](/formulas), and test yourself with timed [practice exams](/practice).

> **Tip:** a partial derivative is just an ordinary derivative with the other variables frozen as constants. Every Calculus 1 rule still works.

## Frequently asked questions

### What is MAC 2313 at UF?

MAC 2313 Analytic Geometry and Calculus 3 is UF's multivariable calculus course. The catalog lists solid analytic geometry, vectors, partial derivatives and multiple integrals.

### What is the prerequisite for MAC 2313?

MAC 2312, MAC 2512 or MAC 3473, with a minimum grade of C.

### Does MAC 2313 cover vector calculus?

The catalog description names solid analytic geometry, vectors, partial derivatives and multiple integrals. Many Calculus 3 courses also introduce line integrals and Green's theorem near the end, so check your section's syllabus for the exact list.

### Is MAC 2313 hard?

The main new skill is thinking in three dimensions, especially when setting up multiple integrals. The computations reuse Calculus 1 and 2, so students with solid integration skills usually adjust quickly.

### How should I prepare for MAC 2313?

Review integration techniques and polar coordinates, then learn vectors, the dot product and the cross product first, since the rest of the course builds on them.