Johns Hopkins AS.110.202 Calculus III: Topics and Practice
A study guide for Johns Hopkins AS.110.202 Calculus III: partial derivatives, multiple integrals, line and surface integrals, Green's and Stokes' theorems.
Johns Hopkins University Updated 3 min read
At a glance
- Course code
- AS.110.202
- Credits
- 4
- Offered
- Not stated in the catalogue entry
- Prerequisites
- C- or better in AS.110.107, 110.109, 110.113, 110.201, 110.212 or 110.302, or a 5 or better on the AP Calculus BC exam
- Covers
- Calculus of functions of more than one variable: partial derivatives and applications, multiple integrals, line and surface integrals, Green's Theorem, Stokes' Theorem and Gauss' Divergence Theorem
- Next course
- Not stated; AS.110.311 Methods of Complex Analysis lists 110.202 as a prerequisite
- Practice here
- Calculus 3
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On this page
Johns Hopkins AS.110.202, Calculus III, is the multivariable calculus course at Hopkins. The e-catalogue describes it in one line: the calculus of functions of more than one variable. This guide unpacks that line into topics and links each one to free practice with instant feedback.
What AS.110.202 covers
According to the Johns Hopkins e-catalogue, AS.110.202 covers partial derivatives and applications; multiple integrals; line and surface integrals; and Green's Theorem, Stokes' Theorem and Gauss' Divergence Theorem. It is a 4-credit course in the Quantitative and Mathematical Sciences distribution area.
The prerequisite is a grade of C- or better in AS.110.107, AS.110.109, AS.110.113, AS.110.201, AS.110.212 or AS.110.302, or a 5 or better on the AP Calculus BC exam. So students arrive by several routes: from either Calculus II, from honors single-variable calculus, or even from linear algebra.
The catalogue does not list vectors separately, but partial derivatives, gradients and surface integrals all rely on them. Expect vectors, lines and planes early in the course.
AS.110.202 topic map
| Calculus III topic | Practice | Read |
|---|---|---|
| Vectors and 3D geometry | Vectors and length | Dot and cross product |
| Functions of several variables | Functions of several variables | Partial derivatives guide |
| Partial derivatives | Partial derivatives | Clairaut's theorem |
| Chain rule | Multivariable chain rule | Chain rule |
| Gradient and tangent planes | Tangent planes | Gradient guide |
| Applications: optimization | Critical points and the second derivative test | Lagrange multipliers |
| Double integrals | Iterated integrals | Double integrals |
| Triple integrals | Triple integrals | Cylindrical and spherical |
| Change of variables | Change of variables | Polar double integrals |
| Vector fields | Vector fields | Conservative fields |
| Line integrals | Line integrals of vector fields | Line integrals of functions |
| Green's Theorem | Green's theorem | Green's theorem |
| Surface integrals | Surface integrals and flux | Curl and divergence |
| Stokes' and Gauss' theorems | Stokes and divergence theorems | Stokes' theorem, Divergence theorem |
AS.110.202 or the honors version?
The catalogue describes AS.110.211 Honors Multivariable Calculus as including the material in AS.110.202 with some additional applications and theory. It is recommended for mathematically able students majoring in the physical sciences, engineering or especially mathematics. If you want the standard computational course, AS.110.202 is the one.
Example problems at the AS.110.202 level
1. A directional derivative. For f(x, y) = x^2 + y^2 at (1, 2) in the direction \mathbf{u} = \langle \frac{3}{5}, \frac{4}{5} \rangle: \nabla f = \langle 2, 4 \rangle, so D_{\mathbf{u}} f = \frac{6}{5} + \frac{16}{5} = \frac{22}{5}.
2. A double integral in polar form.
3. The Divergence Theorem. For \mathbf{F} = \langle x, y, z \rangle over the unit ball, \nabla \cdot \mathbf{F} = 3, so the outward flux is 3 \cdot \frac{4\pi}{3} = 4\pi.
How to do well in AS.110.202
- Build intuition in 3D. Sketch level curves and surfaces regularly, since every later topic depends on picturing them.
- Practice the setup, not just the arithmetic. Most errors in multiple integrals come from limits, so work through switching the order.
- See the theorems as one family. Green's, Stokes' and Gauss' theorems all relate a boundary integral to an interior integral.
- Review Calculus II integration. Techniques from AS.110.109 appear inside every iterated integral.
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Partial derivatives
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