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Berkeley Math 53 Multivariable Calculus: Study Guide, Practice

A study guide for UC Berkeley Math 53: parametric and polar curves, vectors, partial derivatives, multiple integrals and vector calculus, with free practice.

UC Berkeley Updated 3 min read

At a glance

Course code
Math 53
Units
4 (three hours of lecture and three of discussion per week)
Offered
Fall and spring
Prerequisites
Math 1A and 1B (now Math 51 and 52) or equivalent
Covers
Parametric equations and polar coordinates, vectors in 2 and 3 dimensions, vector functions, partial derivatives, multiple integrals, vector calculus with the theorems of Green, Gauss and Stokes, and examples of partial differential equations
Related course
Math 54 (Linear Algebra and Differential Equations)
Practice here
Calculus 3, plus parametric and polar lessons in Calculus 2
On this page
  1. What Berkeley Math 53 covers
  2. Math 53 topic map
  3. Math 53 vs Math 54
  4. Example problems at the Math 53 level
  5. How to do well in Berkeley Math 53

UC Berkeley Math 53 is the multivariable calculus course at Berkeley, taken after the single-variable sequence of Math 51 and Math 52, formerly Math 1A and 1B. This guide maps every Math 53 topic to free practice with instant feedback.

What Berkeley Math 53 covers

According to the Berkeley math department, Math 53 covers parametric equations and polar coordinates, vectors in two and three-dimensional Euclidean space, partial derivatives, multiple integrals, and vector calculus, including the theorems of Green, Gauss and Stokes. The official outline also includes a short unit on vector functions and a final hour on examples of partial differential equations. Vector calculus receives the largest share of class time, followed by multiple integrals and partial derivatives.

That is a standard Calculus 3 course, with the parametric equations and polar coordinates unit that many schools place in Calculus 2 moved to the start.

Math 53 topic map

Math 53 topic Practice Read
Parametric equations Parametric curves Parametric arc length
Polar coordinates Polar coordinates Area in polar coordinates
Vectors in 2 and 3 dimensions Vectors and their length Dot and cross product
Lines and planes Lines in space Planes in space
Vector functions Curves and motion Chain rule
Partial derivatives Partial derivatives Partial derivatives guide
Gradient and tangent planes Tangent planes Gradient and directional derivatives
Maxima, minima and Lagrange multipliers Lagrange multipliers Lagrange multipliers guide
Multiple integrals Double integrals over general regions Double integrals guide
Triple integrals and coordinate systems Cylindrical and spherical Change of variables
Line integrals and Green's theorem Green's theorem Green's theorem
Surface integrals, Stokes and Gauss Stokes and divergence theorems Stokes' theorem, divergence theorem

Math 53 vs Math 54

Course Units Prerequisite Content
Math 53 Multivariable Calculus 4 1A-1B (51-52) or equivalent Curves, vectors, partial derivatives, multiple integrals, vector calculus
Math 54 Linear Algebra and Differential Equations 4 1A-1B, 10A-10B or equivalent Matrices, vector spaces, eigenvalues, linear ODEs and systems, Fourier series

Both are offered in fall and spring. If your single-variable skills are rusty, review the Math 52 guide first.

Example problems at the Math 53 level

1. Area inside a polar curve. The area inside r = 2\cos\theta is

\frac{1}{2}\int_{-\pi/2}^{\pi/2} 4\cos^2\theta\,d\theta = \pi

which matches a circle of radius 1.

2. A tangent plane. For z = x^2 + y^2 at (1, 2, 5), z_x = 2 and z_y = 4, so

z = 5 + 2(x - 1) + 4(y - 2)

3. A triple integral in spherical coordinates. The volume of the ball of radius a:

\int_0^{2\pi}\int_0^{\pi}\int_0^a \rho^2\sin\phi\,d\rho\,d\phi\,d\theta = 2\pi \cdot 2 \cdot \frac{a^3}{3} = \frac{4}{3}\pi a^3

How to do well in Berkeley Math 53

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Points and distance in space

Solve real problems on this topic with instant feedback, hints and a step-by-step walkthrough for every answer. Free, no sign-up needed to start.

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Frequently asked questions

What is Berkeley Math 53?
Math 53, Multivariable Calculus, covers parametric equations and polar coordinates, vectors in two and three dimensions, partial derivatives, multiple integrals and vector calculus, including the theorems of Green, Gauss and Stokes.

What do I need before Math 53?
The department lists Math 1A and 1B, now numbered Math 51 and 52, or equivalent.

Should I take Math 53 or Math 54 first?
Both require the single-variable sequence, so either order is possible. Math 53 is multivariable calculus, while Math 54 covers linear algebra and differential equations.

Is Math 53 hard?
Most of the computations reduce to single-variable calculus, but the course asks you to visualize curves, surfaces and regions in three dimensions. Students who sketch often and practice setups tend to do well.

Written by the PracticeCalculus team

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