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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 and Vectors
3.1
Polar Coordinates
3.2
Curvilinear Coordinates
3.3
Change of Coordinates
3.4
Vectors
3.5
Bases
3.6
Unit Vectors
3.7
The Dot Product
3.8
Visualizing the Dot Product
3.9
The Law of Cosines
3.10
Addition Formulas
3.11
Orthonormal Basis Vectors
3.12
Polar basis vectors
3.13
Orthonormality of Basis Vectors
3.14
The Position Vector
3.15
The Position Vector in Curvilinear Coordinates
3.16
The Position Vector as a Vector Field
3.17
The Distance Formula
3.18
Scalar Fields
3.19
Vector Fields
3.20
The Cross Product
3.21
Lines and Planes
3.22
Linearity of the Dot and Cross Products
4
Potentials due to Discrete Sources
4.1
Electrostatic and Gravitational Potentials and Potential Energies
4.2
Superposition from Discrete Sources
4.3
Visualization of Potentials
4.4
Using Technology to Visualize Potentials
4.5
Two Point Charges
4.6
Series Expansions for Two Point Charges
4.7
Using Technology to Visualize Series Expansions
5
Differentials
5.1
Review of Single Variable Differentiation
5.2
Derivative Notation
5.3
Thick Derivatives
5.4
Differentials
5.5
Rules for Differentials
5.6
Properties of Differentials
5.7
Substitution
5.8
Differentials: Summary
5.9
The Multivariable Differential
6
The Vector Differential
6.1
The Vector Differential
\(d\rr\)
6.2
Finding
\(d\rr\)
on Rectangular Paths
6.3
Other Coordinate Systems
6.4
Calculating Infinitesimal Distance in Cylindrical and Spherical Coordinates
6.5
Calculating
\(d\rr\)
in Curvilinear Coordinates
6.6
Scalar Surface Elements
6.7
Triple Integrals in Cylindrical and Spherical Coordinates
6.8
Using
\(d\rr\)
on More General Paths
6.9
Use What You Know
7
Integration
7.1
Review of Single Variable Integration
7.1.1
Theory
7.1.2
Practice
7.1.3
Skills Check
7.2
Scalar Line Integrals
7.3
Vector Line Integrals
7.4
General Surface Elements
7.5
Vector Surface Elements
7.6
Scalar Surface Integrals
7.7
Volume Integrals
8
Step and Delta Functions
8.1
Kronecker Delta
8.2
Step and Delta Function Motivation
8.3
Step Functions
8.4
The Dirac Delta Function
8.5
Relationship between Delta and Step Functions
8.6
Compare Kronecker and Dirac Deltas
8.7
Dimensions of Step and Delta Functions
8.8
Properties of the Dirac Delta Function
8.9
Representations of the Dirac Delta Function
8.10
The Dirac Delta Function in Three Dimensions
8.11
The Exponential Representation of the Dirac Delta Function
9
Potentials due to Continuous Sources
9.1
Densities
9.2
Densities with Step Functions
9.3
Total Charge
9.4
The Dirac Delta Function and Densities
9.5
Potentials from Continuous Charge Distributions
9.6
Potential Due to a Uniformly Charged Ring
9.7
Potential due to a Finite Line of Charge
9.8
Potential due to an Infinite Line of Charge
10
Gradient
10.1
The Geometry of Gradient
10.2
The Gradient in Rectangular Coordinates
10.3
Properties of the Gradient
10.4
Visualizing the Geometry of the Gradient
10.5
Using Technology to Visualize the Gradient
10.6
Contour Diagrams
10.7
Directional Derivatives
10.8
The Gradient in Curvilinear Coordinates
11
Electric Fields
11.1
The Lorentz Force Law
11.2
Electric Field
11.3
Superposition for the Electric Field
11.4
The Geometry of Electric Fields
11.5
Using Technology to Visualize the Electric Field
11.6
Electric Fields from Continuous Charge Distributions
11.7
Electric Field Due to a Uniformly Charged Ring
11.8
The electric field of a uniform disk
12
Vector Surface Integrals
12.1
Flux
12.2
Dot Products and Components
12.3
Highly Symmetric Surfaces
12.4
Less Symmetric Surfaces
13
Gauss’s Law (Integral Form)
13.1
Flux of the Electric Field
13.2
Gauss’ Law
13.3
Flux through a cube
13.4
Gauss’s Law and Symmetry
13.5
Activity: Gauss’s Law on Cylinders and Spheres
13.6
Electric Field Lines
14
Derivatives of Vector Fields
14.1
The Definition of Divergence
14.2
The Divergence in Two Dimensions
14.3
Exploring the Divergence
14.4
The Divergence in Curvilinear Coordinates
14.5
Exploring the Divergence in Polar Coordinates
14.6
Visualizing Divergence
14.7
The Divergence Theorem
14.8
The Geometry of Curl
14.9
The Definition of Curl
14.10
Exploring the Curl
14.11
The Curl in Curvilinear Coordinates
14.12
Exploring the Curl in Polar Coordinates
14.13
Visualizing Curl
14.14
Stokes’ Theorem
14.15
Second derivatives
14.16
The Laplacian
15
Gauss’s Law (Differential Form)
15.1
Differential Form of Gauss’ Law
15.2
The Divergence of a Coulomb Field
16
Conservative Fields and Energy
16.1
The Relationship between
\(V\text{,}\)
\(\EE\text{,}\)
\(U\text{,}\)
and
\(\FF\)
16.2
Electrostatic Energy from Discrete Charges
16.3
Electrostatic Energy from a Continuous Source
16.4
Independence of Path
16.5
Conservative Vector Fields
16.6
Visualizing Conservative Vector Fields
16.7
Finding Potential Functions
16.8
Finding the Potential from the Electric Field
16.9
Curl-Free Vector Fields
16.10
Divergence-Free Vector Fields
16.11
Second derivatives and Maxwell’s Equations
17
Current, Magnetic Potentials, and Magnetic Fields
17.1
Currents
17.2
Magnetic Vector Potential
17.3
The Biot–Savart Law
17.4
The Magnetic Field of a Straight Wire
17.5
Magnetic Field of a Spinning Ring
17.6
Comparing
\(\boldsymbol{\vec{B}}\)
and
\(\boldsymbol{\vec{A}}\)
for the spinning ring.
18
Ampère’s Law
18.1
Ampère’s Law
18.2
Current in a wire
18.3
Ampère’s Law and Symmetry
18.4
Activity: Ampère’s Law on Cylinders
18.5
Differential Form of Ampère’s Law
19
Conductors and Boundary Conditions
19.1
Conductors
19.2
Boundary conditions on electric fields
19.3
Boundary conditions on magnetic fields
20
Review
20.1
The Relationship between
\(\EE\text{,}\)
\(V\text{,}\)
and
\(\rho\)
20.2
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
Notation
Bibliography
Sections from Other Books
Index
Section
17.6
Comparing
\(\boldsymbol{\vec{B}}\)
and
\(\boldsymbol{\vec{A}}\)
for the spinning ring.
*** Steal results from HW. **
