# UH MATH 2415 Calculus III: Syllabus, Topics and Practice

> A study guide for University of Houston MATH 2415 Calculus III (formerly MATH 2433): vectors, partial derivatives, multiple and line integrals.

Source: https://practicecalculus.com/courses/university-of-houston-math-2415
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 2415 (previously MATH 2433)
- **Credits:** Not printed on the course page; the Texas course number 2415 indicates 4 semester credit hours
- **Prerequisites:** MATH 2414
- **Covers:** Vectors and vector-valued functions, partial differentiation, Lagrange multipliers, multiple integrals, Jacobians, line integrals, Green's Theorem, the Divergence Theorem and Stokes' Theorem
- **Previous course:** MATH 2414 Calculus II
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 2415 is Calculus III at the University of Houston; it was previously numbered MATH 2433.
- It covers vectors and vector-valued functions, partial derivatives, gradients, extreme values and Lagrange multipliers, and double and triple integrals.
- It finishes with line and surface integrals, Green's Theorem, the Divergence Theorem and Stokes' Theorem.
- The prerequisite is MATH 2414.

**UH MATH 2415** (Calculus III) is the third course in the University of Houston calculus sequence and was previously numbered MATH 2433. It lists MATH 2414 as its prerequisite. This guide maps the department's published syllabus to lessons you can practice with instant feedback.

## What UH MATH 2415 covers

The UH Department of Mathematics course guideline describes MATH 2415 as the calculus of functions of several variables: calculus of vector-valued functions, partial differentiation, multiple integrals. Its curriculum details add vectors and vector-valued functions, Lagrange multipliers, Jacobians, and line integrals with Green's Theorem, the Divergence Theorem and Stokes' Theorem. The learning outcomes also mention curvature, velocity, acceleration and torsion for vector-valued functions.

The syllabus is organized in six chapters:

1. **Vectors:** Cartesian space coordinates, vectors, the dot and cross product, lines and planes.
2. **Vector calculus:** vector functions, differentiation formulas, curves, arc length, curvilinear motion and curvature.
3. **Functions of several variables:** graphs, level curves and surfaces, limits and continuity, partial derivatives.
4. **Gradients, extreme values, differentials:** the gradient and directional derivatives, chain rules, tangent planes, local and absolute extremes, Lagrange multipliers, reconstructing a function from its gradient.
5. **Double and triple integrals:** repeated integrals, polar, cylindrical and spherical coordinates, Jacobians and change of variables.
6. **Line and surface integrals:** the Fundamental Theorem for line integrals, Green's Theorem, parameterized surfaces, surface integrals, the Divergence Theorem and Stokes's Theorem.

## MATH 2415 topic map

| MATH 2415 topic | Practice | Read |
| --- | --- | --- |
| Space coordinates and vectors | [Points and distance](/learn/calculus-3/vectors-and-space/points-and-distance) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Dot and cross product | [Cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot product](/learn/calculus-3/vectors-and-space/dot-product) |
| 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) |
| Vector functions, motion, curvature | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric motion](/learn/calculus-2/power-series-parametric-polar/parametric-motion) |
| Level curves and partial derivatives | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Chain rules and tangent planes | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) |
| Extreme values and Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals, polar form | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals](/blog/double-integrals) |
| Triple integrals, cylindrical, spherical | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) |
| Jacobians | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) | [Double integral applications](/learn/calculus-3/multiple-integrals/double-integral-applications) |
| Line integrals and potentials | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Line integrals of functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions) |
| Green's Theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals, Divergence and Stokes | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Divergence theorem](/theorems/divergence-theorem), [Stokes' theorem](/theorems/stokes-theorem) |

## MATH 2415 in the UH sequence

| Course | Old number | Prerequisite |
| --- | --- | --- |
| [MATH 2413 Calculus I](/courses/university-of-houston-math-2413) | MATH 1431 | MATH 2312 with C- or placement |
| [MATH 2414 Calculus II](/courses/university-of-houston-math-2414) | MATH 1432 | MATH 2413 with C- |
| MATH 2415 Calculus III | MATH 2433 | MATH 2414 |

## Example problems at the MATH 2415 level

**1. A tangent plane.** For $z = x^2 + y^2$ at $(1, 1, 2)$:

$$z = 2 + 2(x - 1) + 2(y - 1)$$

**2. A Jacobian.** For polar coordinates $x = r\cos\theta$, $y = r\sin\theta$, the Jacobian is $r$, which is why $dA = r\,dr\,d\theta$.

**3. Green's Theorem.** For $\mathbf{F} = \langle -y, x \rangle$ around the unit circle:

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

## How to do well in MATH 2415

- **Master vectors in the first chapter.** Dot and cross products show up again in tangent planes, flux and Stokes' Theorem.
- **Sketch level curves and regions** before computing anything.
- **Choose coordinates to match the shape** in double and triple integrals.
- **Make a one-page theorem sheet** with what each of Green's, Stokes' and the Divergence Theorem says and when it applies, then test yourself with [vector calculus practice](/learn/calculus-3/vector-calculus/vector-calculus-practice).
- **Use timed [practice exams](/practice)** before each test.

> **Tip:** before parameterizing a line integral, check if the field is a gradient. The syllabus covers reconstructing a function from its gradient for exactly this reason.

## Frequently asked questions

### What is MATH 2415 at the University of Houston?

MATH 2415 Calculus III covers the calculus of functions of several variables: calculus of vector-valued functions, partial differentiation and multiple integrals. The curriculum details add Lagrange multipliers, Jacobians, line integrals, Green's Theorem, the Divergence Theorem and Stokes' Theorem.

### Is MATH 2415 the same as MATH 2433?

Yes. The UH math department says MATH 2415 was previously MATH 2433.

### What is the prerequisite for MATH 2415?

MATH 2414 Calculus II.

### Does MATH 2415 cover curvature and torsion?

Yes. The learning outcomes include calculus on vector-valued functions such as curvature, displacement, velocity, acceleration and torsion.

### What is the hardest part of MATH 2415?

Most students find the last chapter, line and surface integrals with the big theorems, the hardest because it combines everything from the course. Practicing setup and parametrization is the key.