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.
Boston University Updated 3 min read
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
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On this page
Boston University CAS MA 225, Multivariate Calculus, is BU's standard third calculus course. It takes the single-variable tools from MA 123 and 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 | Dot and cross product |
| Lines | Lines in space | Projections and work |
| Planes | Planes in space | Cross product |
| Multiple integration | Iterated integrals | Double integrals |
| Cylindrical and spherical coordinates | Cylindrical and spherical | Triple integrals |
| Partial derivatives | Partial derivatives | Partial derivatives guide |
| Directional derivatives and the gradient | Gradient and directional derivatives | Gradient guide |
| Scalar and vector fields | Vector fields | Functions of several variables |
| Potentials | Conservative fields | Line integrals of vector fields |
| Approximation | Tangent planes | Linear approximation |
| Multivariate minimization | Critical points and the second derivative test | Lagrange multipliers |
| Green's theorem | Green's theorem | Green's theorem |
| Stokes's and related theorems | Stokes and divergence theorems | Stokes' theorem, 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 appear inside every iterated integral.
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Vectors and their length
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