March 2001
PURDUE UNIVERSITY
Study Guide for the Advanced Placement Exam in Multivariate Calculus
This study guide describes briefly the topics to be mastered prior to attempting to obtain credit by
examination for MA 261. The material covered is the calculus of several variables, and it can be studied
from many textbooks, almost all of them entitled CALCULUS or CALCULUS WITH ANALYTIC GEOME-
TRY. The textbook currently used at Purdue is CALCULUS – Early Transcendentals, 4th edition, Stewart,
Brooks/Cole.
IMPORTANT
:
1. Study all the material thoroughly.
2. Solve a large number of exercises.
3. When you feel prepared for the examination, solve the practice problems.
4. Come to the examination rested and confident.
The subject matter of the calculus of several variables extends the student’s ability to analyze functions
of one variable to functions of two or more variables. Graphs of functions of two variables or of equations
involving three variables may be thought of as surfaces in three dimensions. Tangents, normals, and tangent
planes to these surfaces are part of the subject matter of the calculus of several variables.
The concept
of volume is defined for three-dimensional solids.
These new concepts require the introduction of partial
derivatives and multiple integrals.
Most of the problems to be solved require the repeated application of ideas and techniques from the
calculus of one variable.
Accordingly, a good grasp of the notions of di
ff
erentiation and integration for
functions of one variable is a necessary prerequisite for the study of the calculus of several variables. Several
of the concepts from plane analytic geometry are also generalized in the course, leading to a brief study
of three dimensional analytic geometry, including such topics as planes and the quadric surfaces (whose
cross sections are conics), and three dimensional coordinate systems, including rectangular, cylindrical and
spherical coordinates, and the relationships among these.
The topics to be studied prior to attempting the attached practice problems are listed below
1. Analytic Geometry of Three Dimensions
Angle between two vectors, scalar product, cross product, planes, lines, surfaces, curves in 3 dimensional
space.
2. Partial Di
ff
erentiation
Functions of several variables, partial derivatives, di
ff
erential of a function of several variables, partial
derivatives of higher order, chain rule, extreme value problems, directional derivatives, gradient, implicit
functions.
3. Multiple Integrals
Double integrals, iterated integrals in rectangular and polar coordinates, applications, surface integrals,
triple integrals in rectangular, cylindrical and spherical coordinates.
4. Line and surface integrals, independence of path, Green’s theorem, divergence theorem.