Columbia MATH UN1201 Calculus III: Topics, Study Guide and Practice

Columbia MATH UN1201 is Calculus III at Columbia University, the first half of the department's two-course multivariable sequence. Students take it after Calculus I and usually Calculus II, and it is unusual in covering complex numbers and Cramer's rule alongside vectors and partial derivatives. This guide maps every UN1201 topic to practice with instant feedback.

What Columbia MATH UN1201 covers

The Columbia bulletin lists vectors in two and three dimensions, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, and the method of Lagrange multipliers.

That is the first half of a standard Calculus 3: vectors, curves in space and differential calculus in several variables. Multiple integrals and vector calculus wait for UN1202. The complex exponential and Cramer's rule are extras you will not find in most Calculus 3 courses.

UN1201 topic map

UN1201, UN1202 and UN1205

UN1202 requires both UN1102 Calculus II and UN1201, so if you started at UN1201 without Calculus II, plan for it before Calculus IV.

Example problems at the UN1201 level

1. A cross product.

2. The complex exponential and a differential equation. For y'' + y = 0, try y = e^{rx}: then r^2 + 1 = 0, so r = \pm i. Using e^{ix} = \cos x + i\sin x, the real solutions are

3. Lagrange multipliers. Maximize f = xy subject to x + y = 10. From \nabla f = \lambda \nabla g: y = \lambda and x = \lambda, so x = y = 5 and the maximum is f = 25.

How to do well in MATH UN1201

• Get vectors right first. Every later topic uses the dot product and cross product, so make them automatic in the first weeks.
• Revisit Taylor series for the complex exponential. Euler's formula falls out of the series for e^x, \sin x and \cos x. Our Taylor series guide covers them.
• Treat partial derivatives as ordinary derivatives with the other variables frozen. The usual chain rule and product rule still apply.
• Keep the formula sheet handy and use timed practice exams before each midterm.

Tip: in a Lagrange multiplier problem, always check the boundary or the ends of the constraint too. The method finds candidates, and you still compare values.