### Office Hours: TW 10-11, room 215.

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

The text that will be used is:

### Advanced Engineering Mathematicsby Dennis G. Zill, Warren S. Wright, Jones & Bartlett Learning.

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.

#### Arguments covered.

• 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
The next homework will be due on Wednesday April 4th. We will have a second midterm  on Tuesday April 24th based on the second part of the class.

Initial proposals for projects for the finals.