Mathematical background

Vector calculus is a prerequisite to the course. However, it is probably worthwhile to provide some review of these ideas. The following notes should prove helpful.

I. Vectors

A. Definition

B. Dot product

C. Cross product

D. Rotations

E. Exercises (required)

II. Derivatives of vectors and vector identities

A. Curves

B. Gradient

C. Divergence

D. Curl

E. Combining derivatives

F. The Levi-Civita tensor (optional)

G. Exercises (required)

III. Integral theorems

A. Integral of the gradient

B. The divergence theorem

C. Stokes’ theorem

D. Exercises (required)

IV. Additional mathematical tools: Summary

A. Cylindrical coordinates

B. Spherical coordinates

C. Rotations

D. Dirac delta function

E. Helmholz theorem

F. Exercises (required)

G. Additional mathematical tools: Detail (not required)

Electrostatics

Techniques for computing time independent electric fields from a variety of sources.

I. The electric field

A. Coulomb’s law and the electric field

B. The continuum limit

C. Exercises (required)

D. Math reminder: Taylor series

II. Gauss's law

A. The integral form of Gauss’s law

B. Examples using the integral form of Gauss's law

C. Exercises (required)

III. Maxwell’s equations for electrostatics

A. The differential form of Gauss’s law

B. The curl of the electric field

C. Maxwell’s equations for electrostatics and the electric potential

D. Boundary conditions

E. Examples

F. Exercises (required)

IV. Electric potential energy

A. Conservation of energy in the presence of electrostatic forces

B. The energy of a charge configuration

C. Force on a charged surface

D. Capacitance

E. Examples

F. Exercises (required)

Midterm exam: Midterm Review

Solution techniques for electrostatics

I. The method of images

A. Planes

B. Spheres

C. Exercises (required)

II. Separation of variables in Cartesian coordinates

A. Separation of variables

B. Satisfying the boundary conditions

C. Exercise (required)

II. Separation of variables in spherical coordinates

A. Separation of variables

B. Satisfying the boundary conditions. These examples show some of the range of applications solvable using the spherical expansion. However, you will only be

required to do problems similar to the exercises.

C. Exercises (required)

III. Multipole expansion

A. The multipole expansion

B. Monopole

C. Dipole

D. Quadrupole (optional)

E. Exercises (required)

Electric fields in matter

I. Electric fields in matter

A. Polarization of molecules

B. The potential produced by polarized materials