Problem 1.1: The plot of the following function looks like a hill on the xy plane:
h(x,y) = exp(2ry) - 3x^2 - 4y - 18x + 28y - 5/60
(a) Where is the top of the hill located?
(b) How high is the hill?
(c) In what direction is the slope steepest at the point (1,1)?
(d) How steep is the slope of h(x,y) at the point (1,1) in the direction n = r + y?
Problem 1.2: The separation vector can be written as R = (x - x) + (y - y) + (z - z). If R = R, what is the magnitude of the separation vector? Show that ∇R is a unit vector parallel to R.
Problem 1.3: Find the scalar function o(x) whose gradient is ∇ = (2ry + xy + 3rz).
Problem 1.4: Evaluate the gradient of the following scalar functions: (i) ∇ = ∇ln(r) and (ii) ∇ = 1/r.
Problem 1.5:
(a) Calculate the divergence ∇ · E of the vector E = i/r, where n is an integer and i is the unit vector corresponding to vector r = xxi + yyi + zzi.
(b) What is ∇ · E when n = 2?
(c) In what physical context are vector-functions of type E = f(r) encountered?
Problem 1.6: Suppose ∇ · E = 0 and ∇ × B = 0. Show that ∇ × E = -∂B/∂t and ∇ · B = 0.
Problem 1.7: Griffiths Problem 1.33 - Verify Stoke's theorem for the function v = ry + 2y = y + 3, using the shaded area shown in Fig. 1(a).
Problem 1.8: Griffiths Problem 1.53 - Verify the divergence theorem for the function v = rcosθ + cosθsinφ using the one octant of the sphere of radius R as the volume (see Fig. 1(b)).
(a)
(b)
FIG. 1:
