Review of vector calculus and and its application to partial
differential equations'

The text that will be used is:

The syllabus can be found here.

There will be two midterm.

The exercise listed are for HW collection. I will collect them
every
two weeks and grade 2 or 3 exercises among the one assigned.

The final grade will be based on the following rules: 45% final, 35% midterms, 20% HW. Curving will be done on the final result.

- Multidimensional Calculus
- Curves and surfaces, gradients, divergence and curl
- Taylor expansions in IR3
- Divergence and Stokes theorem
- Classification of partial differential equations
- The concept of well-posed problems

- Potential Problems
- Derivation of Laplace's equation; Dirichlet and Neumann problems
- The maximum principle and uniqueness of solutions
- Green's identities and Green's functions for selected domains
- Connections to variational problems and complex variables

- Parabolic Problems
- Derivation of the heat equation in IR3; discussion of boundary and initial conditions; the maximum principle for the heat equation and uniqueness of solutions; fundamental solution for pure initial value problems; Duhamel's principle for inhomogeneous equations

- Hyperbolic Problems
- The concept of characteristics for a single first order equation
- Solution of initial value problems; the concept of a shock
- D'Alembert solution of the wave equation; Huyghen's principle and the solution of the wave equation in IR3

**First week**

Material covered:

- 9.1 Vector Function

- 9.2 Motion on a curve

- 9.3 Curvature and component of the acceleration

Exercises:

- (9.1): 19, 26, 31, 36

- (9.2): 6, 8, 17, 20

- (9.3): 4, 21, 22

**Second week**

Material covered:

- 9.4 Partial Derivatives

- 9.5 Directional Derivative

- 9.6 Tangent Planes and Normal Lines

Exercises:

- (9.4): 33, 43, 55

- (9.5): 29, 39, 40

- (9.6): 21, 22, 38

**Third week**

Material covered:

- 9.7 Curl and Divergence

- 9.8 Line Integrals

- 9.9 Independence of the Path

Exercises:

- (9.7): 36, 43, 44

- (9.8): 30, 37, 39

- (9.9): 17, 18, 21

**Fourth week**

Material covered:

- 9.10 Double Integrals

- 9.11 Double Integrals in Polar Coordinates

- 9.15 Triple Integrals

Exercises:

- (9.10):

- (9.11):

- (9.15):

**Fifth week**

Material covered:

- 9.12 Green’s Theorem

- 9.13 Surface Integrals

- 9.14 Stokes’ Theorem

Exercises:

- (9.12): 8, 12, 27, 28

- (9.13): 13, 18, 43

- (9.14): 8, 15, 18

**Sixth week**

Material covered:

- 12.2 Fourier Series

- 12.3 Fourier Sine and Cosine Series

Exercises:

- (12.2): 4, 5, 10, 14

- (12.3): 5, 6, 8, 29, 31

**Seventh week**

Material covered:

- 12.2 Complex Fourier Series

- 13.1 Separable Differential Equation

Exercises:

- (12.4): 3, 6 ,7

- (13.1): 3, 6, 7, 8, 12

**Eighth week**

Material covered:

- 13.2 Classical PDEs and Boundary Value Problem

- 13.3 Heat Equation

Exercises:

- (13.2): 1, 3, 5, 8, 10

- (13.3): 4, 6, 7

**Nineth week**

Material covered:

- 13.4 Wave Equation

- 13.5 Potential Equation

Exercises:

- (13.4): 4, 7, 9, 12

- (13.5): 3, 7, 16, 17

Initial proposals for projects for the finals.