# Boston University MA 225 Multivariate Calculus: Topics and Practice

> A study guide for Boston University CAS MA 225 Multivariate Calculus: vectors, multiple integrals, partial derivatives, gradients and Stokes's theorem.

Source: https://practicecalculus.com/courses/boston-university-ma-225
Updated: 2026-11-08

## At a glance

- **Course code:** CAS MA 225
- **Units:** 4
- **Offered:** Fall and spring (sections listed for Fall 2026 and Spring 2027)
- **Prerequisites:** CAS MA 124 or CAS MA 129
- **Covers:** Vectors, lines, planes; multiple integration; cylindrical and spherical coordinates; partial and directional derivatives; scalar and vector fields; the gradient; potentials; approximation; multivariate minimization; Stokes's and related theorems
- **Next course:** Not stated on the course page
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- CAS MA 225 is Boston University's multivariate calculus course, a 4-unit course offered in fall and spring.
- It covers vectors, lines and planes, multiple integration in several coordinate systems, partial derivatives, gradients, potentials and minimization.
- It ends with Stokes's theorem and related theorems.
- It cannot be taken for credit in addition to CAS MA 230, the honors-level vector calculus course.

**Boston University CAS MA 225**, Multivariate Calculus, is BU's standard third calculus course. It takes the single-variable tools from [MA 123](/courses/boston-university-ma-123) and [MA 124](/courses/boston-university-ma-124) into two and three dimensions and finishes with Stokes's theorem. This guide maps each topic on the course page to free practice with instant feedback.

## What MA 225 covers

The BU course page lists: vectors, lines, planes; multiple integration, cylindrical and spherical coordinates; partial derivatives, directional derivatives, scalar and vector fields, the gradient, potentials, approximation, multivariate minimization; and Stokes's and related theorems. The prerequisite is CAS MA 124 or MA 129, and the course carries 4 units.

MA 225 cannot be taken for credit in addition to CAS MA 230. It fulfills one BU Hub unit each in Quantitative Reasoning II and Critical Thinking. Sections are listed for Fall 2026 and Spring 2027, and students register for a discussion section alongside the lecture.

## MA 225 topic map

| MA 225 topic | Practice | Read |
| --- | --- | --- |
| Vectors | [Vectors and length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) | [Projections and work](/learn/calculus-3/vectors-and-space/projections-and-work) |
| Planes | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Cross product](/learn/calculus-3/vectors-and-space/cross-product) |
| Multiple integration | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives guide](/blog/partial-derivatives) |
| Directional derivatives and the gradient | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Scalar and vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) |
| Potentials | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) |
| Approximation | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Multivariate minimization | [Critical points and the second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Stokes's and related theorems | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [Divergence theorem](/theorems/divergence-theorem) |

## MA 225 or MA 230?

BU offers two versions of this course, and you can earn credit for only one.

| Course | What it adds |
| --- | --- |
| CAS MA 225 Multivariate Calculus | The standard computational course described above |
| CAS MA 230 Honors-Level Vector Calculus | Linear algebra (linear transformations, matrices, inner products, cross products), tangent spaces, flows, divergence and curl, and the integral theorems with more theory |

Both list MA 124 or MA 129 as the prerequisite.

## Example problems at the MA 225 level

**1. Distance from a point to a plane.** The distance from $(1, 2, 3)$ to $x + 2y + 2z = 0$ is $\frac{|1 + 4 + 6|}{\sqrt{1 + 4 + 4}} = \frac{11}{3}$.

**2. A potential.** $\mathbf{F} = \langle y, x + 2y \rangle$ has potential $f = xy + y^2$, since $f_x = y$ and $f_y = x + 2y$.

**3. Minimization.** For $f(x, y) = x^2 + xy + y^2 - 3x$, solving $2x + y = 3$ and $x + 2y = 0$ gives a minimum at $(2, -1)$.

## How to do well in MA 225

- **Start with solid vector skills.** Lines, planes and the dot and cross products support everything that follows.
- **Draw regions before integrating.** Choosing the right coordinate system, Cartesian, cylindrical or spherical, depends on the shape.
- **Test for a potential first.** If a field is conservative, a hard line integral becomes a simple difference of two values.
- **Review single-variable integration.** Methods from [MA 124](/courses/boston-university-ma-124) appear inside every iterated integral.

## Frequently asked questions

### What is BU CAS MA 225?

CAS MA 225 is Multivariate Calculus at Boston University: vectors, lines and planes, multiple integration, cylindrical and spherical coordinates, partial and directional derivatives, scalar and vector fields, the gradient, potentials, approximation, multivariate minimization, and Stokes's and related theorems.

### What are the prerequisites for MA 225?

CAS MA 124 or CAS MA 129.

### What is the difference between MA 225 and MA 230?

MA 230, Honors-Level Vector Calculus, adds linear algebra (linear transformations, matrices, inner products) and treats differentiation and integration with more theory. Students cannot get credit for both.

### Which BU Hub areas does MA 225 fulfill?

Since Fall 2020, one unit each in Quantitative Reasoning II and Critical Thinking.

### What does "potentials" mean in MA 225?

A potential is a scalar function whose gradient equals a given vector field. When a field has one, its line integrals depend only on the endpoints.