Math 15, Winter 2006,
Mathematics for the physical sciences
Covered topics
- 1. Functions of several variables
- Matrices
- Derivatives and linear approximation (IMPS 6.1)
- Chain-rule (IMPS 6.2)
- Square matrices, inverses and determinants (IMPS 6.3)
- Integration (IMPS 1.5)
- Change of variables (IMPS 6.4)
- Application: Charge and charge density; Mass (IMPS 1.1-1.5)
- 2. Vector fields
- Flow lines (IMPS 1.8, 2.2)
- Line integrals (DGV 63)
- Path-independence and convervative vector fields (DGV 71)
- Grad, Curl and potential functions (DGV 75, 115)
- Application: Gravity; Electric fields and electric potential (IMPS 1.6-1.11)
- 3. Flux and circulation
- Flux in 2 dimensions (IMPS 2.2-2.3)
- Divergence and the divergence theorem (IMPS 2.7)
- Circulation and Green's theorem (IMPS 4.8)
- Application: Fluid flow; sources and sinks (IMPS 2.1-2.3)
- Surfaces and surface integrals (IMPS 2.5, 3.3, DGV Chapter 2)
- Divergence theorem revisited (IMPS 2.7, DGV Chapter 2)
- Stokes' theorem (IMPS 4.9, DGV 93)
- Application: Electric flux and Gauss's Law (IMPS 2.4-2.8)
- Everything revisited
- Application: Magnetic fields and Induction (IMPS 4)
- 4. Optimization
- 2nd derivatives, critical points and local optimization (IMPS 5.1-5.3)
- Constrained optimization; Lagrange multipliers (IMPS 5.4)
- 5. Taylor Series (IMPS 7)