Chabot College
Fall 2004
Replaced Fall 2010
Course Outline for Mathematics 3
MULTIVARIABLE CALCULUS
Catalog Description:
3 - Multivariable Calculus
5 units
Vector valued functions, functions of several variables, partial differentiation, multiple integration, change of
variables theorem, scalar and vector fields, gradient, divergence, curl, line integral, surface integral,
Theorems of Green, Stokes and Gauss, applications. Prerequisite: Mathematics 2 (completed with a
grade of "C" or higher). 5 hours.
[Typical contact hours: 87.5]
Prerequisite Skills:
Before entering the course, the student should be able to:
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define the natural logarithmic function in terms of a Riemann integral;
integrate and differentiate logarithmic functions;
define and differentiate inverse functions;
define an exponential function;
differentiate and integrate exponential functions;
differentiate and integrate inverse trigonometric functions;
differentiate and integrate hyperbolic functions and their inverses;
solve application problems involving logarithmic, exponential, inverse trigonometric, and hyperbolic
functions;
solve differential equations using separation of variables;
use standard techniques of integration such as integration by parts, trigonometric integrals, trigonometric
substitution, partial fractions, rational functions of sine and cosine;
graph polar equations and find area of regions enclosed by the graphs of polar equations;
evaluate limits using L'Hopital's Rule;
evaluate improper integrals;
use parametric representations of plane curves;
perform basic vector algebra in R2 and R3 and interpret the results geometrically;
find equations of lines and planes in R3;
construct polynomial approximations (Taylor polynomials) for various functions and estimate their
accuracy using an appropriate form of the remainder term in Taylor's formula;
determine convergence of sequences;
determine whether a series converges absolutely, converges conditionally or diverges;
construct (directly or indirectly) power series representations (Taylor series) for various functions,
determine their radii of convergence, and use them to approximate function values.
Expected Outcomes for Students:
Upon completion of the course, the student should be able to:
1. perform vector algebra in R3, interpret the results geometrically especially the dot and cross products and
apply them to problems in space geometry (eg. orthogonal projection, vector equations of lines and
planes);
2. transform points and equations among rectangular, cylindrical, and spherical coordinates and sketch
their graphs as well as quadric surfaces;
3. parameterize curves using vector functions of one variable and analyze them (eg. find unit tangent, unit
normal, curvature);
4. sketch the graphs of functions of two variables using level curves, traces in coordinate planes, symmetry,
etc.;
5. extend the concepts of limits, continuity, differentiability and differential of single variable functions to
Chabot College
Course Outline for Math 3, page 2
Fall 2004
functions of two variables;
6. compute limits, partial derivatives, total differential, gradient, directional derivatives and interpret them
geometrically and in terms of rate of change;
7. apply partial derivatives and/or gradients to problems involving tangent planes and linear approximation,
and optimization, especially using Lagrange multipliers;
8. compute double and triple integrals directly or using change of variables and explain the geometric
interpretation of Jacobians;
9. apply differential operators gradient, divergence, curl and Laplacian to scalar and vector fields and
interpret the results;
10. compute line integrals using parameterizations for curves;
11. parameterize surfaces using vector functions of two variables, and compute their areas;
12. compute surface integrals of scalar functions and vector functions using parameterization for surfaces;
13. interpret the theorems of Green, Stokes and Gauss physically as well as mathematically (as the
generalizations of the Fundamental Theorem of Calculus), and use them to compute line and surface
integrals;
14. find scalar potentials for conservative vector field.
Course Content:
1. Space Analytic Geometry
a. Rectangular, cylindrical and spherical coordinate systems
b. Surfaces (cylinders, quadric surfaces and surfaces of revolution)
c. Vector algebra and geometry in R3 especially dot and cross product, lines and planes
d. Vector functions and parameterized curves
2. Partial Differentiation
a. Scalar functions of two or more variables
b. Limits, continuity, partial derivatives, Chain Rule
c. Tangent planes and linear approximation
d. Optimization (especially using Lagrange multipliers)
3. Multiple Integration
a. Double and triple integrals and their evaluations in rectangular, polar, cylindrical and
b. Spherical coordinates
c. Change of variables and Jacobians
d. Area, volume, surface area, center of mass and moments of inertia
4. Scalar and Vector Fields
a. Vector differential operators (gradient, divergence, curl and Laplacian)
b. Identities involving differential operators
5. Integrals over Paths and Surfaces
a. Integral of scalar functions along a path and line integrals
b. Path independence
c. Parameterized surface and its area
d. Surface integral
6. Vector Analysis
a. Green's and Stokes' Theorems
b. Scalar potentials and conservative fields
c. Gauss' Theorem
Methods of Presentation:
1. Lectures
2. Discussions
Chabot College
Course Outline for Math 3, page 3
Fall 2004
Assignments and Methods of Evaluating Student Progress:
1. Typical Assignments
a. Let f be a function of two variables that has continuous partial derivatives and consider the points
A(1, 3), B(3, 3), C(1, 7), and D(6, 15). The directional derivative of f at A in the direction of the vector
AB is 3 and the directional derivative at A in the direction of AC is 26. Find the directional derivative
of f at A in the direction of the vector AD.
b. Two legs of a right triangle are measured as 5 m and 12 m with a possible error in measurement of
at most 0.2 cm in each. Use differentials to estimate the maximum error in the calculated value of
(a) the area of the triangle and (b) the length of the hypotenuse.
2. Methods of Evaluating Student Progress
a. Homework
b. Quizzes
c. Exams and final exam
Textbook(s) (Typical):
Calculus, James Stewart, Brook/Cole, 2003
Special Student Materials:
None.
Revised 10/20/03