Ohio State MATH 2153 Calculus III: Topics, Study Guide, Practice

Ohio State MATH 2153, Calculus III, is the multivariable calculus course at Ohio State. It is a 4-credit course offered in autumn, spring and summer, and it is the next step after either MATH 1152 or the engineering course MATH 1172. This guide maps every MATH 2153 topic to practice with instant feedback.

What Ohio State MATH 2153 covers

The catalog describes MATH 2153 simply as multivariable differential and integral calculus. The department's course description document lists four chapters of topics.

Vectors and vector-valued functions: vectors in the plane and in space, dot and cross products, lines and curves in space, calculus of vector-valued functions, motion, arc length and curvature. Functions of several variables: planes and surfaces, level curves, limits and continuity, partial derivatives, the chain rule, directional derivatives and the gradient, tangent planes, max/min problems and Lagrange multipliers. Multiple integration: double integrals over rectangles and general regions, polar coordinates, triple integrals in cylindrical and spherical coordinates, mass calculations and change of variables. Vector calculus: vector fields, line integrals, conservative fields, Green's theorem, divergence and curl, surface integrals, Stokes' theorem and the Divergence Theorem.

That is a complete Calculus 3 course.

MATH 2153 topic map

How students reach MATH 2153

Students coming from MATH 1172 have already seen partial derivatives and the gradient. Students coming from MATH 1152 have seen more series and polar coordinates. See the MATH 1152 guide and the MATH 1172 guide to fill any gaps.

Example problems at the MATH 2153 level

1. A tangent plane. For z = x^2 + y^2 at (1, 1, 2): z_x = 2x = 2 and z_y = 2y = 2, so the tangent plane is

2. Curl of a vector field. For \mathbf{F} = \langle -y, x, 0 \rangle,

3. A triple integral. The volume of the unit cube is \int_0^1\int_0^1\int_0^1 1\,dz\,dy\,dx = 1. With the density \rho = z, the mass is

How to do well in Ohio State MATH 2153

• Sketch every region and surface. Wrong bounds are the most common lost points. Practice switching the order to build that skill.
• Learn the coordinate systems together. Polar, cylindrical and spherical each add a factor to the volume element. The formula sheet lists them side by side.
• See the big picture of vector calculus. Read Green's theorem first, then Stokes' theorem and the Divergence theorem as generalizations.
• Test yourself with timed practice exams, and skim Calculus 1 vs 2 vs 3 to see how the courses connect.

Tip: for a conservative field, a line integral depends only on the endpoints. Checking for a potential function first can save a page of work.