# Georgia Tech MATH 2551 Multivariable Calculus: Study Guide, Practice

> A study guide for Georgia Tech MATH 2551 (Multivariable Calculus): linear approximation, Lagrange multipliers, multiple integrals and vector analysis.

Source: https://practicecalculus.com/courses/georgia-tech-math-2551
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 2551
- **Credits:** 4 credit hours
- **Covers:** Linear approximation and Taylor's theorems, Lagrange multipliers and constrained optimization, multiple integration, and vector analysis including the theorems of Green, Gauss and Stokes
- **Credit restrictions:** No credit for both MATH 2551 and MATH 2401, MATH 2411 or MATH 2561
- **Related course:** MATH 2550 (Introduction to Multivariable Calculus, 2 credit hours, no vector calculus)
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 2551 is Georgia Tech's 4-credit-hour multivariable calculus course.
- It covers linear approximation and Taylor's theorems, Lagrange multipliers, multiple integration and vector analysis.
- The course ends with the big theorems of vector calculus: Green's, Gauss's (the divergence theorem) and Stokes'.
- MATH 2550 is a 2-credit-hour introduction to multivariable calculus that leaves out vector calculus.

**Georgia Tech MATH 2551** is Multivariable Calculus, a 4-credit-hour course that takes calculus into several dimensions and finishes with vector analysis. It builds on single-variable differential and integral calculus (MATH 1551 and MATH 1552 at Georgia Tech). This guide maps every MATH 2551 topic to free practice, so you can learn each idea by solving problems with instant feedback.

## What MATH 2551 covers

The Georgia Tech catalog describes MATH 2551 as covering linear approximation and Taylor's theorems, Lagrange multipliers and constrained optimization, multiple integration, and vector analysis including the theorems of Green, Gauss and Stokes. Credit is not awarded for both MATH 2551 and MATH 2401, MATH 2411 or MATH 2561.

That is a full standard **Calculus 3** course. The phrase "linear approximation" covers tangent planes, differentials and the gradient, while vector analysis brings in vector fields, line and surface integrals, and the three integral theorems that tie them together.

## MATH 2551 topic map

| MATH 2551 topic | Practice | Read |
| --- | --- | --- |
| Vectors and geometry in space | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [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) |
| Linear approximation | [Tangent planes and linear approximation](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Taylor's theorems | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor's theorem](/theorems/taylors-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) |
| Unconstrained optimization | [Second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization problems](/blog/optimization-problems) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double and triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integrals](/blog/double-integrals) |
| Change of variables | [Change of variables and the Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [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) |
| Gauss and Stokes | [Stokes' theorem and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Divergence theorem](/theorems/divergence-theorem), [Stokes' theorem](/theorems/stokes-theorem) |

## MATH 2551 vs MATH 2550

| Course | Credit hours | Covers | Vector calculus |
| --- | --- | --- | --- |
| MATH 2550 (Introduction to Multivariable Calculus) | 2 | Vectors in 3D, space curves, functions of several variables, partial derivatives, optimization, multiple integration | Not covered |
| MATH 2551 (Multivariable Calculus) | 4 | Linear approximation, Taylor's theorems, constrained optimization, multiple integration, vector analysis | Green, Gauss and Stokes |

The catalog lists MATH 2550 and MATH 2551 as mutually exclusive for credit. For the single-variable background, see the [MATH 1551 guide](/courses/georgia-tech-math-1551) and the [MATH 1552 guide](/courses/georgia-tech-math-1552).

## Example problems at the MATH 2551 level

**1. A linear approximation.** For $f(x, y) = \sqrt{x^2 + y^2}$ at $(3, 4)$, $f = 5$, $f_x = \frac{3}{5}$ and $f_y = \frac{4}{5}$, so

$$f(3.1, 3.9) \approx 5 + \frac{3}{5}(0.1) + \frac{4}{5}(-0.1) = 4.98$$

**2. Lagrange multipliers.** Maximize $f = xy$ subject to $x + y = 10$. From $\nabla f = \lambda \nabla g$: $y = \lambda$ and $x = \lambda$, so $x = y = 5$ and the maximum is $f(5, 5) = 25$.

**3. Green's theorem.** For the counterclockwise unit circle $C$ with $P = -y$ and $Q = x$, $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2$, so

$$\oint_C -y\,dx + x\,dy = \iint_D 2\,dA = 2\pi$$

## How to do well in Georgia Tech MATH 2551

- **Sketch every region.** Integration limits are where most points are lost. Practice reading regions with [general regions](/learn/calculus-3/multiple-integrals/general-regions) and [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order).
- **Tie the gradient to geometry.** It points uphill fastest and is perpendicular to level curves, which is the whole idea behind Lagrange multipliers. See the [gradient guide](/blog/gradient-and-directional-derivatives).
- **Sort the integral theorems.** Closed plane curve: Green. Closed surface: Gauss (divergence). Surface with a boundary curve: Stokes. Drill them in [vector calculus practice](/learn/calculus-3/vector-calculus/vector-calculus-practice).
- **Keep single-variable skills fresh** with the [formula sheet](/formulas), and rehearse with timed [practice exams](/practice).

> **Tip:** before computing a line integral directly, check whether the field is conservative. If it is, the [potential function](/learn/calculus-3/vector-calculus/conservative-fields) turns the integral into a simple subtraction.

## Frequently asked questions

### What is MATH 2551 at Georgia Tech?

MATH 2551 is Multivariable Calculus, a 4-credit-hour course. The catalog lists linear approximation and Taylor's theorems, Lagrange multipliers and constrained optimization, multiple integration and vector analysis, including the theorems of Green, Gauss and Stokes.

### What is the difference between MATH 2551 and MATH 2550?

MATH 2550, Introduction to Multivariable Calculus, is 2 credit hours and covers vectors in three dimensions, curves in space, functions of several variables, partial derivatives, optimization and multiple integration. It does not cover vector calculus, which MATH 2551 does.

### Is MATH 2551 hard?

The main new skill is thinking in three dimensions: picturing surfaces, setting up limits for triple integrals and choosing coordinates. Students with solid single-variable integration usually find the computations familiar once the setup is right.

### How should I prepare for MATH 2551?

Review integration techniques and the chain rule from single-variable calculus, then preview vectors, the dot and cross product, and partial derivatives. These appear in the first weeks and everything else builds on them.