IU MATH-M 311 Calculus 3: Syllabus, Topics and Practice

IU MATH-M 311 (Calculus 3) is the multivariable calculus course at Indiana University Bloomington and the third course after MATH-M 211 and MATH-M 212. IU lists it at 3 to 5 credits. This guide maps each topic in the official description to lessons you can practice with instant feedback.

What IU MATH-M 311 covers

IU's official course listing describes MATH-M 311 as: elementary geometry of 2, 3, and n-space, functions of several variables, partial differentiation, minimum and maximum problems, multiple integration.

That covers the core of Calculus 3: vectors and the geometry of lines and planes, functions of two or more variables, partial derivatives and gradients, optimization including constrained problems, and double and triple integrals. The listing does not mention line integrals, surface integrals or Green's and Stokes' theorems, so treat those as extras unless your syllabus includes them. The listing also does not give a textbook or schedule.

MATH-M 311 topic map

M311 in the IU Bloomington sequence

How M311 builds on M211 and M212

Nearly every M311 skill reuses one-variable calculus. A partial derivative is an ordinary derivative with the other variables held fixed, and a double integral is two single integrals done one after the other. That makes it worth reviewing derivative rules and integration techniques before the term starts.

Example problems at the M311 level

1. Distance in 3-space. Between (1, 2, 3) and (4, 6, 3):

2. A constrained maximum. Maximize f = xy subject to x + y = 10. Lagrange multipliers give y = \lambda and x = \lambda, so x = y = 5 and the maximum is 25.

3. A double integral over a triangle.

How to do well in MATH-M 311

• Think geometrically. Picture vectors as arrows and functions of two variables as surfaces.
• Treat other variables as constants when taking a partial derivative. Practice with partial derivatives practice.
• Draw the region first for every double or triple integral, then choose the order of integration.
• Practice Lagrange multipliers problems and double integral problems.
• Use timed practice exams and the IU Math Learning Center before tests.

Tip: at a critical point of f(x, y), use D = f_{xx}f_{yy} - f_{xy}^2. If D > 0 and f_{xx} > 0 it is a minimum, if D > 0 and f_{xx} < 0 a maximum, and if D < 0 a saddle.