# UGA MATH 2270 Calculus III: Syllabus, Topics and Practice

> A study guide for UGA MATH 2270 Calculus III for Science and Engineering: gradients, Lagrange multipliers, multiple integrals and vector calculus.

Source: https://practicecalculus.com/courses/university-of-georgia-math-2270
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 2270
- **Credit hours:** 4
- **Prerequisites:** MATH 2260, MATH 2310H, MATH 2410 or MATH 2410H
- **Covers:** Parametric curves and planetary motion, derivatives, the gradient, Lagrange multipliers, multiple integration, polar, cylindrical and spherical coordinates, line and surface integrals, Green's, Stokes's and Divergence theorems
- **Duplicate credit:** Not open to students with credit in MATH 2500, MATH 3510 or MATH 3510H
- **Offered:** Fall, spring and summer every year
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 2270 is the 4-credit-hour Calculus III for Science and Engineering course at the University of Georgia.
- It covers functions of two and three variables: parametric curves, the gradient, Lagrange multipliers, and multiple integrals in polar, cylindrical and spherical coordinates.
- It ends with line and surface integrals and Green's, Stokes's and the Divergence theorems, with physics applications.
- The prerequisite is MATH 2260, MATH 2310H, MATH 2410 or MATH 2410H.

**UGA MATH 2270** (Calculus III for Science and Engineering) is the third course in the University of Georgia's science and engineering calculus sequence. It is a 4-credit-hour course offered every fall, spring and summer. This guide maps the official topic list to lessons you can practice with instant feedback.

## What UGA MATH 2270 covers

The UGA Department of Mathematics describes MATH 2270 as the calculus of functions of two and three variables: parametric curves and applications to planetary motion. Derivatives, the gradient, Lagrange multipliers. Multiple integration, area, volume, and physical applications, polar, cylindrical, and spherical coordinates. Line and surface integrals, Green's, Stokes's, and Divergence theorems, with applications to physics.

That is a full **Calculus 3** course. Because UGA introduces vectors, dot and cross products, and lines and planes in MATH 2260, MATH 2270 can move quickly into curves, partial derivatives, multiple integrals and vector calculus. The department page does not list a schedule or textbook; it links a syllabi lookup.

## MATH 2270 topic map

| MATH 2270 topic | Practice | Read |
| --- | --- | --- |
| Vector review | [Vectors and length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Parametric curves and motion | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric motion](/learn/calculus-2/power-series-parametric-polar/parametric-motion) |
| Functions of two and three variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) |
| The gradient | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Multiple integration, area, volume | [General regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Physical applications | [Double integral applications](/learn/calculus-3/multiple-integrals/double-integral-applications) | [Center of mass](/learn/calculus-2/improper-integrals-and-applications/center-of-mass) |
| Polar coordinates | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) |
| Stokes's and Divergence theorems | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [Divergence theorem](/theorems/divergence-theorem) |

## MATH 2270 in the UGA sequence

| Course | Credit hours | Prerequisite |
| --- | --- | --- |
| [MATH 2250 Calculus I for Science and Engineering](/courses/university-of-georgia-math-2250) | 4 | MATH 1113 or department permission |
| [MATH 2260 Calculus II for Science and Engineering](/courses/university-of-georgia-math-2260) | 4 | MATH 2250 or MATH 2300H |
| MATH 2270 Calculus III for Science and Engineering | 4 | MATH 2260, 2310H, 2410 or 2410H |

## Example problems at the MATH 2270 level

**1. Velocity and speed.** For $\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$:

$$\mathbf{v}(t) = \langle -\sin t, \cos t, 1 \rangle, \qquad |\mathbf{v}(t)| = \sqrt{2}$$

**2. A spherical integral.** The volume of a ball of radius $R$:

$$\int_0^{2\pi}\int_0^{\pi}\int_0^R \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta = \frac{4}{3}\pi R^3$$

**3. The Divergence Theorem.** For $\mathbf{F} = \langle x, y, z \rangle$ over the unit ball, $\nabla\cdot\mathbf{F} = 3$, so the outward flux is $3 \cdot \frac{4}{3}\pi = 4\pi$.

## How to do well in MATH 2270

- **Review the MATH 2260 vectors unit first.** Dot and cross products drive everything from tangent planes to flux.
- **Sketch regions and solids** before setting up any multiple integral.
- **Memorize the volume elements:** $r\,dr\,d\theta$ in polar and cylindrical, $\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$ in spherical.
- **Compare the big theorems side by side** using the [vector calculus summary](/learn/calculus-3/vector-calculus/vector-calculus-summary).
- **Practice with [Lagrange multipliers problems](/practice-problems/lagrange-multipliers)** and timed [practice exams](/practice).

> **Tip:** if a surface is closed and the divergence is simple, the Divergence Theorem usually turns a hard flux integral into an easy volume integral.

## Frequently asked questions

### What is MATH 2270 at UGA?

MATH 2270 Calculus III for Science and Engineering covers the calculus of functions of two and three variables: parametric curves and applications to planetary motion, derivatives, the gradient, Lagrange multipliers, multiple integration, area, volume and physical applications, polar, cylindrical and spherical coordinates, and line and surface integrals with Green's, Stokes's and Divergence theorems.

### What is the prerequisite for MATH 2270?

MATH 2260, MATH 2310H, MATH 2410 or MATH 2410H.

### Can I take both MATH 2270 and MATH 2500?

No. MATH 2270 is not open to students with credit in MATH 2500, MATH 3510 or MATH 3510H.

### Does MATH 2270 start with vectors?

At UGA, vectors in three-dimensional space, dot and cross products, and lines and planes are part of MATH 2260. MATH 2270 builds on them, so review that unit before the term starts.

### What does planetary motion have to do with MATH 2270?

The course uses parametric curves in space to describe motion, and the department names planetary motion as an application. Velocity and acceleration vectors along a curve are the key tools.