1 Introduction 
Page References 3
How to Use this Book 3

Plotting Functions

2 Graphical Facilities 5
Downloading the FlexPDE Program 
The FlexPDE Editor 
Plotting Functions of One Variable 
Saving and Running the Descriptor File 
Plotting Functions of Two Variables 
Running FlexPDE from the Explorer 
Help and Manual 

3 "Curly" Velocity Fields 16
Liquid Rotating as a Disk 
Non-Constant w  
More General Velocity Fields 

4 Fields of Gravitation 25
Earth and Moon as Point Masses 
Planets of Finite Size 
Divergence of g  

Electricity

5 Fields around Electric Charges 34
Field around Two Positive Point Charges 
Positive and Negative Point Charges 
The Dipole Field 
Field around Charged Wires 
Dipole of Charged Wires 
The Gauss Integral 
Positive and Negative Charges in (r,z) 

6 Laplace and Poisson Equations 48
Known Values on Boundaries 
Values and Derivatives on Boundaries 
Derivative Boundary Conditions Only 
Solution Over a Quarter-Circle 
The Poisson Equation 

7 Electrostatic Fields in (x,y) Space 56
Metal Rod in a Metal Box 
Metal Bar in a Tube 
Metal Rod across a Parallel Field 

8 Electrical Conduction in (x, y) Space 63
Conduction in a Rectangular Plate 
Conduction in a Trapezoidal Plate 
Checking the Solution 
Radial Conduction in a Foil 
Constricted Rectangular Plate 
Plate Made of Two Different Metals 
Plate with an Elliptic Insert 

9 Dielectrics in (x, y) Space 76
Coaxial Cable 
Capacitance 
Parallel Plate Capacitor 
Glass Rod across a Parallel Field 
Surface Charge of Polarization 

10 Steady Fields in (r , z) Space 90
Electrical Conduction in a Cone 
Plate Capacitor 
Glass Ball in a Parallel Field 
Charged, Conducting Ellipsoid 
Spherical Capacitor 
Charged Dielectric 

11 Electrostatic Force and Energy 108
Forces on a Parallel-Plate Capacitor 
Oppositely Charged Metal Rods 
Charged Metal Balls 
Electrostatic Energy 
Field Energy of Metal Balls in Glass 
Energy Minimum (Optional) 
Forces on a Dielectric 
Metal and Dielectric Rod across a Parallel Field 
Metal Ball and Dielectric Ellipsoid in (r,z) 

12 Magnetostatics in (x, y) Space 124
Magnetic Field around a Wire 
Field around Two Wires 
Force between Two Wires 
Two Wires inside a Magnetic Shield 
Boundary Conditions for Magnetic Vectors 
Force on a Magnetic Object 
Force on a Magnetic Cylinder 
Model of a Permanent Magnet 

13 Magnetostatics in (r,z) Space 141
Magnetic Field around a Wire 
Field along the (r, z) Plane 
Simple Magnet Coil 
Helmholtz Coil 
Forces on Coils 
Coil with a Magnetic Core 
Force between a Coil and a Magnetic Ellipsoid 
Magnetostatic Field Energy 
Simple Model of a Permanent Magnet 
Permanent Magnetization 
Permanent Magnet as a Current Sheet 

Heat

14 Heat Conduction in (x, y) Space 162
Two Hot-Water Tubes 
Uniformly Heated, Semi-Circular Rod 
Conduction in Anisotropic Wood (Optional) 

15 Heat Conduction in (r, z) Space 172
Radial Conduction in a Hollow Cylinder
Cooling by Forced Convection 
Continuously Varying Thermal Conductivity 
Steel Tube With Cooling Flanges 

16 Non-Linear Conduction 180
Temperature Dependent Conductivity 
Steel Foil Emitting Infrared Radiation 

17 Simultaneous Electric and Thermal Conduction 186
Copper Block heated by an Electric Current 
Electrically heated, radiating Copper Foil 
Semicircular Foil, heated by an Electric Current 

18 Transient Problems in One Space Dimension 195
Iron Bar with a Temperature Step 
Bar with a Temperature Step Half-Way 
Iron Bar, Soldered to a Copper Bar 
Ramp Function at one Boundary 

19 Transients in (x, y) and (r, z) Space 204
Internally heated Steel Bar with a Loss 
Capsule and Sample in a Scanning Calorimeter 
Moving Heat Source 
Stationary State for a Moving Heat Source 

20 Time-Sinusoidal Problems 213
Oscillating Temperature in a Steel Block 
Animation of an Oscillating Temperature 
Sinusoidal Volume Heating of a Steel Foil 

Metal and Semiconductor Physics

21 Electron and Hole Conduction 222
Steady-State Electron Current 
One-Dimensional Conduction in a Metal 
Field at a Metal-Vacuum Interface 
Semiconductors 
Semiconductors in One Dimension 
Semiconductor in an Electric Field 
Current through a Semiconductor 
Model of a Silicon Junction Diode 
Reverse Current

Three Dimensional Problems

22 Fields in Three Dimensions 240
Two Point Charges of Different Signs
Non-Linear Set of Point Charges
Laplace Equation in 3D
Coin in a Metal Box
Electrical Conduction in a Cone
Exploiting Symmetry
Glass Block in a Parallel Electric Field
Simple Magnet Coil
Steady Heat Conduction
Temperature Transients

Appendix

Principles of Finite Element Analysis 268
Interpolation and Differentiation in 2D
Interpolating to Obtain a Solution
Solving for Node Values
Natural Boundary Conditions

Conclusion 275

References 276

Vocabulary 277