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THE GEOMETRY OF STATIC FIELDS
Corinne A. Manogue, Tevian Dray
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Front Matter
Colophon
Acknowledgments
Notation
1
Introduction
1.1
Philosophy
1.2
Learning Outcomes
2
Static Vector Fields — Prerequisites
2.1
Dimensions
2.2
Voltmeters
2.3
Computer Algebra
3
Coordinates, Vectors, and Fields
3.1
Visualizing Polar Coordinates
3.2
Polar and Rectangular Coordinates
3.3
Curvilinear Coordinates
3.4
Change of Coordinates
3.5
Vectors
3.6
Bases
3.7
Unit Vectors
3.8
The Dot Product
3.9
Visualizing the Dot Product
3.10
The Law of Cosines
3.11
Addition Formulas
3.12
Polar basis vectors
3.13
Curvilinear Basis Vectors
3.14
Orthonormality of Basis Vectors
3.15
The Position Vector
3.16
The Position Vector in Curvilinear Coordinates
3.17
The Position Vector as a Vector Field
3.18
The Distance Formula
3.19
Scalar Fields
3.20
Vector Fields
3.21
Vector Fields
3.22
The Cross Product
3.23
Lines and Planes
3.24
Linearity of the Dot and Cross Products
4
Power Series
4.1
Definition of Power Series
4.2
Calculating Power Series Coefficients
4.3
Visualization of Power Series Approximations
4.4
Discussion of Approximations using Power Series
4.5
Using Technology to Explore Power Series Approximations
4.6
Guessing Power Series from Graphs
4.7
Common Power Series
4.8
Dimensions in Power Series
4.9
Convergence of Power Series
4.10
Theorems about Power Series
5
Potentials due to Discrete Sources
5.1
Electrostatic and Gravitational Potentials and Potential Energies
5.2
Superposition from Discrete Sources
5.3
Visualization of Potentials
5.4
Using Technology to Visualize Potentials
5.5
Two Point Charges
5.6
Series Expansions for Two Point Charges
5.7
Multipole Expansions (Power vs. Laurent Series)
5.8
Using Technology to Visualize Series Expansions
6
Differentials
6.1
Review of Single Variable Differentiation
6.2
Derivative Notation
6.3
Thick Derivatives
6.4
Differentials
6.5
Rules for Differentials
6.6
Properties of Differentials
6.7
Substitution
6.8
Differentials: Summary
6.9
The Multivariable Differential
7
The Vector Differential
7.1
The Vector Differential
\(d\rr\)
7.2
Finding
\(d\rr\)
on Rectangular Paths
7.3
Other Coordinate Systems
7.4
Calculating Infinitesimal Distance in Cylindrical and Spherical Coordinates
7.5
Calculating
\(d\rr\)
in Curvilinear Coordinates
7.6
Scalar Surface Elements
7.7
Triple Integrals in Cylindrical and Spherical Coordinates
7.8
Using
\(d\rr\)
on More General Paths
7.9
Use What You Know
8
Integration
8.1
Review of Single Variable Integration
8.1.1
Theory
8.1.2
Practice
8.1.3
Skills Check
8.2
Scalar Line Integrals
8.3
Vector Line Integrals
8.4
General Surface Elements
8.5
Vector Surface Elements
8.6
Scalar Surface Integrals
8.7
Volume Integrals
9
Step and Delta Functions
9.1
Kronecker Delta
9.2
Step and Delta Function Motivation
9.3
Step Functions
9.4
The Dirac Delta Function
9.5
Relationship between Delta and Step Functions
9.6
Compare Kronecker and Dirac Deltas
9.7
Dimensions of Step and Delta Functions
9.8
Properties of the Dirac Delta Function
9.9
Representations of the Dirac Delta Function
9.10
The Dirac Delta Function in Three Dimensions
9.11
The Exponential Representation of the Dirac Delta Function
10
Potentials due to Continuous Sources
10.1
Densities
10.2
Densities with Step Functions
10.3
Total Charge
10.4
The Dirac Delta Function and Densities
10.5
Potentials from Continuous Charge Distributions
10.6
Potential Due to a Uniformly Charged Ring
10.7
Potential due to a Finite Line of Charge
10.8
Potential due to an Infinite Line of Charge
11
Gradient
11.1
The Geometry of Gradient
11.2
The Gradient in Rectangular Coordinates
11.3
Properties of the Gradient
11.4
Visualizing the Geometry of the Gradient
11.5
Using Technology to Visualize the Gradient
11.6
Contour Diagrams
11.7
Directional Derivatives
11.8
The Gradient in Curvilinear Coordinates
12
Electric Fields
12.1
The Lorentz Force Law
12.2
Electric Field
12.3
Superposition for the Electric Field
12.4
The Geometry of Electric Fields
12.5
Using Technology to Visualize the Electric Field
12.6
Electric Fields from Continuous Charge Distributions
12.7
Electric Field Due to a Uniformly Charged Ring
12.8
The electric field of a uniform disk
13
Vector Surface Integrals
13.1
Flux
13.2
Dot Products and Components
13.3
Highly Symmetric Surfaces
13.4
Less Symmetric Surfaces
14
Gauss’s Law (Integral Form)
14.1
Flux of the Electric Field
14.2
Gauss’ Law
14.3
Flux through a cube
14.4
Gauss’s Law and Symmetry
14.5
Activity: Gauss’s Law on Cylinders and Spheres
14.6
Electric Field Lines
15
Derivatives of Vector Fields
15.1
The Definition of Divergence
15.2
The Divergence in Two Dimensions
15.3
Exploring the Divergence
15.4
The Divergence in Curvilinear Coordinates
15.5
Exploring the Divergence in Polar Coordinates
15.6
Visualizing Divergence
15.7
The Divergence Theorem
15.8
The Geometry of Curl
15.9
The Definition of Curl
15.10
Exploring the Curl
15.11
The Curl in Curvilinear Coordinates
15.12
Exploring the Curl in Polar Coordinates
15.13
Visualizing Curl
15.14
Stokes’ Theorem
15.15
Second derivatives
15.16
The Laplacian
16
Gauss’s Law (Differential Form)
16.1
Differential Form of Gauss’ Law
16.2
The Divergence of a Coulomb Field
17
Conservative Fields and Energy
17.1
The Relationship between
\(V\text{,}\)
\(\EE\text{,}\)
\(U\text{,}\)
and
\(\FF\)
17.2
Electrostatic Energy from Discrete Charges
17.3
Electrostatic Energy from a Continuous Source
17.4
Independence of Path
17.5
Conservative Vector Fields
17.6
Visualizing Conservative Vector Fields
17.7
Finding Potential Functions
17.8
Finding the Potential from the Electric Field
17.9
Curl-Free Vector Fields
17.10
Divergence-Free Vector Fields
17.11
Second derivatives and Maxwell’s Equations
18
Current, Magnetic Potentials, and Magnetic Fields
18.1
Currents
18.2
Magnetic Vector Potential
18.3
The Biot–Savart Law
18.4
The Magnetic Field of a Straight Wire
18.5
Magnetic Field of a Spinning Ring
18.6
Comparing
\(\boldsymbol{\vec{B}}\)
and
\(\boldsymbol{\vec{A}}\)
for the spinning ring.
19
Ampère’s Law
19.1
Ampère’s Law
19.2
Current in a wire
19.3
Ampère’s Law and Symmetry
19.4
Activity: Ampère’s Law on Cylinders
19.5
Differential Form of Ampère’s Law
20
Conductors and Boundary Conditions
20.1
Conductors
20.2
Boundary conditions on electric fields
20.3
Boundary conditions on magnetic fields
21
Review
21.1
The Relationship between
\(\EE\text{,}\)
\(V\text{,}\)
and
\(\rho\)
21.2
The Relationship between
\(\boldsymbol{\vec{B}}\text{,}\)
\(\boldsymbol{\vec{A}}\text{,}\)
and
\(\boldsymbol{\vec{J}}\)
Back Matter
A
Formulas
A.1
Formulas for Div, Grad, Curl
A.1.1
Rectangular Coordinates
A.1.2
Cylindrical Coordinates
A.1.3
Spherical Coordinates
A.2
Product Rules
A.3
Integration by parts
B
Symbols
C
Notation
Bibliography
Sections from Other Books
Index
Section
20.1
Conductors
Please read Section 2.5 in Griffiths.