| I. |
Multivariable Functions (f : (x,y) → R,
F : (x,y,z) → R)
|
A. | Graphs, Contours
(level curves, level surfaces) |
| B. | Limits & Continuity
|
1. | Definitions (delta-epsilon, distance, uniqueness) |
| 2. | Computing/Showing Nonexistence
(simplify, polar coordinates, substitution, differing paths) |
|
| | C. | Differentiation
|
1. | Partial Derivatives (definition, computing, interpretation:
slope of tangent line,
rate of change) |
| 2. | Tangent Planes (unique tangent planes, approximations) |
| 3. | Tangent Plane ⇔
Differentiable ⇔ Smooth |
| 4. | Jacobian Matrix, Jacobian Determinant |
| 5. | Basic Rules (linearity, product) |
| 6. | Implicit Partial Differentiation,
Implicit Function Theorem |
| 7. | Higher-Order & Mixed Partial Derivatives |
| | D. | Multivariable Chain Rule |
| E. | Directional Derivatives |
| F. | Gradient
| | G. | Extrema
(local & absolute maximum & minimum, pringle (saddle) points) |
|
| II. |
3-D Basics
|
A. | Coordinate Systems
(right-handed, Cartesian, cylinderical, spherical)
| | B. | Distance |
| C. | 3-D Graphics |
| D. | Vectors in 3-D
|
1. | Length |
| 2. | Dot Product (definition, cos representation,
dot product equals zero implies perpendicular) |
| 3. | Cross Product (definition, sin representation,
cross product equals zero implies parallel) |
|
|
| III. |
Vector Functions (definitions: r: t --> Rn
)
|
A. | Vector Differentiation and Integration
| | B. | Lines |
| C.
| Particle Motion: velocity, acceleration,
(tangential vector, T(t),
arc length, s(t),
speed, s′(t)
= ||r′(t)||
= ||v(t)|| ,
v(t) = s′(t)T(t)
)
|
|
| IV. |
Multiple Riemann Integrals (double and triple integrals)
| A. | Definition (Riemann Sums) |
| B. | Iterated Integrals (Fubini theorem) |
| C. | Double Integrals
(area, volume, mass, center of mass)
|
| D. | Integration in Polar Coordinates
(change of variables, Jacobian determinant) |
| E. | Triple Integrals
(volume, mass, center of mass) |
| F. | Integration
in Cylindrical
and Spherical
Coordinates
(change of variables, Jacobian determinant) |
|
