# Northwestern MATH 230-1 Multivariable Differential Calculus

> A study guide for Northwestern MATH 230-1 Multivariable Differential Calculus: vectors, vector functions, partial derivatives and optimization, with practice.

Source: https://practicecalculus.com/courses/northwestern-math-230-1
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 230-1
- **Units:** 1 unit
- **Offered:** Not stated on the First-Year Focus page
- **Prerequisites:** MATH 218-3, MATH 214-0 (former), MATH 220-2 or MATH 224-0 (former); not open to students in the McCormick School of Engineering
- **Covers:** Vectors, vector functions, partial derivatives and optimization
- **Next course:** MATH 230-2 (Multivariable Integral Calculus)
- **Practice here:** [Calculus 3](/learn/calculus-3), first two units

## Key takeaways

- MATH 230-1 is the first half of Northwestern's multivariable calculus sequence for students outside the engineering school.
- It covers vectors, vector functions, partial derivatives and optimization.
- Multiple integrals and vector calculus are in the second half, MATH 230-2.
- McCormick engineering students take MATH 228-1 instead.

**Northwestern MATH 230-1**, Multivariable Differential Calculus, is the first half of Northwestern's multivariable calculus sequence for students outside the McCormick School of Engineering. Like the single-variable sequence, it splits differentiation and integration across two quarters. MATH 230-1 handles the differential side. This guide maps each topic to free practice with instant feedback.

## What MATH 230-1 covers

Northwestern's course description is brief: vectors, vector functions, partial derivatives, and optimization. The course is worth one unit. The prerequisite is MATH 218-3, MATH 214-0 (former), MATH 220-2 or MATH 224-0 (former), and the course is not open to students in the McCormick School of Engineering, who take MATH 228-1 instead.

In practice, "vectors" means vector algebra and the geometry of lines and planes in space; "vector functions" means curves in space and motion; "partial derivatives" covers the multivariable chain rule, gradients, directional derivatives and tangent planes; and "optimization" covers critical points and constrained optimization. That matches the first half of a typical **Calculus 3** course. Multiple integrals and vector calculus follow in MATH 230-2.

## MATH 230-1 topic map

| MATH 230-1 topic | Practice | Read |
| --- | --- | --- |
| Points and distance in 3D | [Points and distance](/learn/calculus-3/vectors-and-space/points-and-distance) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vectors | [Vectors and length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot product](/learn/calculus-3/vectors-and-space/dot-product) |
| Cross product | [Cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Projections and work](/learn/calculus-3/vectors-and-space/projections-and-work) |
| 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 | [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 several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives guide](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Chain rule | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Tangent planes | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Optimization | [Critical points and the second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Partial derivatives practice](/practice-problems/partial-derivatives) |
| Constrained optimization | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |

## Placement and the sequence around 230-1

The department's First-Year Focus page recommends MATH 230-1 for students with a 5 on AP Calculus AB or a 4 or 5 on AP Calculus BC, listing MATH 290-1 and MATH 226-0 as alternatives for those same scores. MATH 226-0, Sequences and Series, is not a prerequisite for multivariable calculus and can be taken at the same time. The standard path is 230-1 followed by MATH 230-2, Multivariable Integral Calculus.

## Example problems at the MATH 230-1 level

**1. A plane through three points.** For $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$, the normal $\langle -1, 1, 0 \rangle \times \langle -1, 0, 1 \rangle = \langle 1, 1, 1 \rangle$ gives the plane $x + y + z = 1$.

**2. Velocity of a vector function.** For $\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$, $\mathbf{r}'(t) = \langle -\sin t, \cos t, 1 \rangle$, with speed $\sqrt{2}$.

**3. Classifying a critical point.** For $f(x, y) = x^2 - y^2$, the only critical point is $(0, 0)$, and $D = (2)(-2) - 0 = -4 < 0$, so it is a saddle point.

## How to do well in MATH 230-1

- **Make vector algebra fast.** Dot and cross products show up in nearly every problem, so drill them early.
- **Visualize with level curves.** Contour plots make gradients and optimization much easier to understand.
- **Connect to single-variable ideas.** Partial derivatives, tangent planes and the second derivative test all extend ideas from [MATH 220-1](/courses/northwestern-math-220-1).
- **Mix problem types before exams.** The [partial derivatives practice](/learn/calculus-3/partial-derivatives/partial-derivatives-practice) set mixes topics the way exams do.

## Frequently asked questions

### What is Northwestern MATH 230-1?

MATH 230-1, Multivariable Differential Calculus, is a one-unit course on vectors, vector functions, partial derivatives and optimization.

### What are the prerequisites for MATH 230-1?

MATH 218-3, MATH 214-0 (former), MATH 220-2 or MATH 224-0 (former). The course is not open to students in the McCormick School of Engineering.

### Which AP scores place into MATH 230-1?

The department lists MATH 230-1 as a recommended placement for a 5 on AP Calculus AB and for a 4 or 5 on AP Calculus BC.

### Do I need MATH 226-0 before MATH 230-1?

No. The department says MATH 226-0 is not a prerequisite for multivariable calculus and can be taken at the same time as MATH 230-1 or 230-2.

### What comes after MATH 230-1?

MATH 230-2, Multivariable Integral Calculus, which covers the integration half of multivariable calculus.