MULTIVARIABLE CALCULUS
XIAOLONG HAN
Abstract.
This lecture note is designed for a onesemester course in Calculus III: Mul
tivariable Calculus. We follow the textbook [S] closely and homework is assigned from
it, too. I also use [A] as an excellent reference.
Introduction
This course covers vectors and multivariable calculus. Topics include vectors and their
calculus in two and threedimensional spaces, parametric curves, partial derivatives and
multiple integrals. There are two general interpretations: geometrical and analytical, the
connection between them lies in every subject in this course, so do not hesitate to exploit
your geometric visualization and analytic computation!
As its name suggests, multivariable calculus is the extension of calculus to more than
one variables. That is, in single variable calculus we study functions of a single independent
variable
y
=
f
(
x
)
.
In multivariable calculus we study functions of two or more independent variables, e.g.
z
=
f
(
x, y
) or
w
=
f
(
x, y, z
)
.
These functions are interesting in their own right, but they are also essential for de
scribing the physical world. We are also interested in vector functions, i.e., the functions
with vector values as outputs.
By the end of the course we will know how to differentiate and integrate functions of
several variables. In single variable calculus the Fundamental Theorem of Calculus relates
derivatives to integrals. We will see something similar in multivariable calculus and the
capstone to the course will be the three theorems (Green’s, Stokes’ and Gauss’) that do
this.
Contents
Introduction
1
1.
Vectors and the geometry of space
2
1.1.
Threedimensional rectangular coordinate systems and distance formula
2
1.2.
Vectors and their basic properties
4
1.3.
The dot product
6
1.4.
The cross product
8
1.5.
Lines and planes in threedimensional space
10
2.
Vector functions
13
1
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XIAOLONG HAN
2.1.
Vector functions and space curves
13
2.2.
Derivatives and integrals of vector functions
14
2.3.
Arc length
16
2.4.
Motion in space: velocity and acceleration
16
3.
Partial derivatives
17
3.1.
Functions of several variables
17
3.2.
Partial derivatives
19
3.3.
Tangent planes and linear approximations
20
3.4.
The chain rule and implicit function theorem
21
3.5.
Directional derivatives and the gradient vector
22
3.6.
Maximum and minimum values
23
4.
Multiple integrals
25
4.1.
Double integrals over rectangles
25
4.2.
Iterated integrals
26
4.3.
Double integrals over general regions
27
4.4.
Double integrals in polar coordinates
28
4.5.
Triple integrals over rectangular boxes
29
4.6.
Change of variables in double integrals
29
References
30
1.
Vectors and the geometry of space
Recall
xy
plane, twodimensional space, as the frame of single variable calculus, we
introduce threedimensional space as the frame of multivariable calculus (strictly, two
variables). The frame includes coordinate systems, vectors and their calculus. Objectives
in this chapter are
(1) Building threedimensional rectangular coordinate systems and deriving distance
formula;
(2) Defining vectors in two and threedimensional spaces and introducing vector cal
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 Spring '09
 Calculus, Derivative, Multivariable Calculus, Dot Product, XIAOLONG HAN

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