Northwestern MATH 230-1 Multivariable Differential Calculus

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

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.
• Mix problem types before exams. The partial derivatives practice set mixes topics the way exams do.