| I. |
Vectors
| A. | Basics
(addition, negation, right-hand coordinates, length, unit vector) |
| B. | Dot Product
| 1. | definition vs. aibi |
| 2. | for perpendicular vectors |
| 3. | component of a
in the direction of b |
| 4. | equation of a plane |
| | C. | Cross Product
| 1. | definition vs. epsilonijkaibj |
| 2. | for parallel vectors |
| 3. | right-hand rule |
| 4. | rotational velocity |
| | D. | Triple Products
|
|
| II. |
Integration
|
A. | Basics (from Calculus II) |
| B. | Line Integrals (work, circulation, conservative)
|
| C. | Surface Integrals
(flux, n dA = rx  ry dydx) |
| D. | Volume Integrals (density) |
|
| III. |
Differential Operators
| A. | Background
|
1. | Grad(f(x,y)) vs. Grad(F(x,y,z)) |
| 2. | Directional Derivative |
| 3. | Level Curve, Level Surface |
| | B. | Potential Functions |
| C. | Divergence, div(u)
|
1. | definition (interpretation) vs. formula |
| 2. | solenoidal, divergence free |
| | D. | Laplcian |
| E. | Curl, curl(u)
|
1. | definition (interpretation) vs. formula |
| 2. | irrotational, curl free |
|
|
| IV. |
Suffix, Index, Tensor Notation
| A. | Basic Rules
|
1. | Free Suffix |
| 2. | Einstein Summation Convention |
| | B. | Special Tensors
(Kronecker Delta, Alternating Tensor) |
| C. | Differential Operators in Index Notation
(definitions, combinations, product rules) |
|
| V. |
Green's Theorems (General Form)
| A. | Divergence Theorem (Gauss's Theorem) |
| B. | Stokes's Theorem
(Curl Theorem) |
| C. | Related Theorems (Green's Identities) |
|
| VI. |
Cartesian Tensors and Rotation
| A. | Coordinate Transforms |
| B. | Rotation Transform Rules for Tensors |
| C. | Symmetric/Antisymmetric Tensors |
| D. | Isotropic Tensors |
|
ry dydx)