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Vector Calculus:
An Introduction
Daniel Daners
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Front Matter
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Colophon
Acknowledgements
Preface
I
Functions of Several Variables
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1
Vectors
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1.1
Euclidean Space
1.2
Addition and Multiplication by Scalars
1.3
Scalar Product and Norm
1.4
The Area of a Parallelogram
1.5
The Cross Product
1.6
The Volume of a Parallelepiped
2
Functions of Several Variables
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2.1
Graphs
2.2
Contour Maps
2.3
Level Sets
2.4
Vector Valued Functions
3
Limits and Continuity
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3.1
Sequences of Vectors
3.2
Open and Closed Sets
3.3
Continuous Functions
4
Partial Derivatives
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4.1
Basic Definitions
4.2
Derivatives of Vector Valued Functions
4.3
The Chain Rule
4.4
Directional Derivatives
4.5
Tangents and Normals
4.5.1
The Direction of Most Rapid Increase
4.5.2
Normals to Level Sets
4.6
Higher Order Derivatives
4.7
Taylor Polynomials
4.8
Maxima and Minima
4.8.1
Critical Points: The First Derivative Test
4.8.2
The Hessian Matrix: The Second Derivative Test
4.9
Maxima and Minima with Constraints
II
Multiple Integrals
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5
Double Integrals
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5.1
Definition of Double Integrals
5.2
Evaluation of Double Integrals
5.3
The Transformation Formula
5.4
Application: Polar Coordinates
6
Integrals of More Variables
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6.1
Definition of Triple Integrals
6.2
Evaluation of Triple Integrals
6.3
The Transformation Formula
6.4
Application: Spherical and Cylindrical Coordinates
6.4.1
Spherical Coordinates
6.4.2
Cylindrical Coordinates
6.5
Comments on More Variables
III
Line and Surface Integrals
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7
Line integrals
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7.1
Basic Definitions
7.2
Arc Length
7.3
Integrals of a Scalar Function
7.4
Integrals of a Vector Field
8
Conservative Vector Fields
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8.1
Gradient Vector Fields and Potentials
8.2
Closed Vector Fields
8.3
Closed Vector Fields in the Plane
8.4
Closed Vector Fields in Space
9
Surface Integrals
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9.1
Parametric Representation of Surfaces
9.2
Implicit and Explicit Representations of Surfaces
9.3
Unit Normals and Orientation
9.4
The Comuptation of Unit Normals
9.4.1
Normals for surfaces defined implicitly
9.4.2
Computation of unit normals using parametric representations
9.5
Surface Integrals
9.5.1
The Definition of Surface Integrals
9.5.2
The Jacobian for spherical coordiates
9.5.3
The Jacobian for a graph
9.6
The Flux Across a Surface
IV
The Integral Theorems of Vector Calculus
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10
Green’s Theorem
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10.1
Domains and Their Boundaries
10.2
Green’s Theorem
10.3
Application: The Area of a Domain.
10.4
Application: Conservative Vector Fields
11
Two Interpretations of Green’s Theorem
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11.1
The Divergence Theorem in the Plane
11.2
Physical Interpretation of the Divergence
11.3
The Theorem of Stokes in the Plane
12
The Divergence Theorem and Greens Identities
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12.1
The Divergence Theorem
12.2
Green’s First and Second Identity
13
The Theorem of Stokes
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13.1
Surfaces with a Boundary
13.2
The Theorem of Stokes
13.3
Application: Conservative Vector Fields in Space
Back Matter
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A
Notation
References
Index
Section
2.1
Graphs
Given a function from a set
D
⊂
R
2
with values in
R
we can plot its value on the
x
3
-axis as shown in
Figure 2.1
. The resulting object is a surface, called the graph of
.
f
.
Formally, it is the set
🔗
graph
(
f
)
:=
{
(
x
1
,
x
2
,
f
(
x
1
,
x
2
)
)
∣
(
x
1
,
x
2
)
∈
D
}
in
.
R
3
.
🔗
🔗
Figure
2.1
.
To obtain the graph of
f
plot
f
(
x
)
above
.
x
∈
D
.
🔗
As an example consider the graph of
f
(
x
,
y
)
:=
5
y
(
1
−
y
2
)
(
1
/
2
−
x
2
)
−
x
in
Figure 2.2
.
🔗
Figure
2.2
.
Graph of
.
f
(
x
,
y
)
:=
5
y
(
1
−
y
2
)
(
1
/
2
−
x
2
)
−
x
.
🔗
If
D
⊂
R
N
we still call the set
🔗
graph
(
f
)
:=
{
(
x
,
f
(
x
)
)
∣
x
∈
D
}
⊂
R
N
+
1
the
graph of
f
over
D
, even if we cannot visualise it!
🔗
🔗
🔗
