Homework 6

1) Math 5.86) a) Find the directional derivative of U = 2xy-z² at (2, -1, 1) in a direction toward (3, 1, -1).
b) In what direction is the directional derivative a maximum?
c) What is that maximum value?

2) Math 6.69) Evaluate ∫[(2x - y + 4)dx + (5y + 3x - 6)dy] around a circle of radius 4 with centre at (0, 0).

3) A charge Q sits at the centre of a cube of side a. Calculate the electric flux through one side of the cube by direct integration and also by Gauss's Law. (Hint, careful use of symmetry will simplify this problem so that you only have to actually do one integral.)

4) Grant & Phillips 2.1 (you may already have done this, if so just hand it it in again).

5) Grant & Phillips 2.2.

6) Grant & Phillips 2.3.

7) Grant & Phillips 2.5.

8) Grant & Phillips 2.6.

9) Here there are 6 pictures of electric fields. Which of them have div E equal to zero and which have non-zero div E? If some have non-zero div E indicate in what regions the non-zero divergence is found.

10) Explain what is meant by the terms "Linear" and "Isotropic" when referring to a dielectric material. What would be different about the behavior of a material which was NOT isotropic compared to a material that was?