Syllabus

SM279

Multivariable Calculus

1. Affine Geometry of ¨ⁿ

a. Lines - Parametric

b. Planes - Parametric

c. Subspaces

Review

i. Linear Independence

ii. Spanning

iii. Bases

iv. Dimension

d. k-dimensional affine sets – parametric

e. Dot Product

f. Implicit form for Affine sets

i. Hyperplanes

ii. Review of Solutions of Equations

g. Convex Sets

h. Open Balls

i. Interior, Exterior and Boundary Points

j. Open and Closed Sets

2. Functions from ¨ⁿ to¨^m.

a. Curves in ¨ⁿ.

b. Surfaces in ¨ⁿ

c. Linear Functions

i. Review of Matrix Operations

d. ¨ⁿ to ¨^m functions

e. Partial Derivatives

f. Derivative Matrix

g. Directional Derivative

3. Chain Rule

a. Chain Rule Theorem

b. Inverse Function Theorem

c. Implicit Function Theorem

4. Generalized Inner Products

a. Change of Coordinates

i. Review Coordinates

b. Perpendicular Subspaces

i. Review Rank-Nullity Theorem

c. Positive Definiteness

i. Review Determinants

ii. Review Gram- Schmidt Process

5. Unconstrained Optimization

a. Second Derivative Test

6. Constrained Optimization

a. Second Derivative Test

7. Systems of Differential Equations

b. Review Eigenvalues/Eigenvectors