Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
Figure 1.23.
(b) Iff is a di€erentiable function and A is a di€erentiable vector function,
then
r…f A† ˆ…rf †A ‡ f …r A†:
1.18. (a) What is the divergence of a gradient?
(b) Show that r 2 …1=r† ˆ0.
(c) Show that r …rr† 6ˆ…rr†r.
1.19 Given rE ˆ 0; rH ˆ 0; rE ˆ@H=@t; rH ˆ @E=@t, show that
E and H satisfy the wave equation r 2 u ˆ @ 2 u=@t 2 .
The given equations are related to the source-free Maxwell's equations of
electromagnetic theory, E and H are the electric ®eld and magnetic ®eld
intensities.
1.20. (a) Find constants a, b, c such that
A ˆ…x 1 ‡ 2x 2 ‡ ax 3 †^e 1 ‡…bx 1 3x 2 x 3 †^e 2 ‡…4x 1 ‡ cx 2 ‡ 2x 3 †^e 3
is irrotational.
(b) Show that A can be expressed as the gradient of a scalar function.
1.21. Show that a cylindrical coordinate system is orthogonal.
1.22. Find the volume element dV in: (a) cylindrical and (b) spherical coordinates.
Hint: The volume element in orthogonal curvilinear coordinates is
dV ˆ h 1 h 2 h 3 du 1 du 2 du 3 ˆ @…x 1; x 2 ; x 3 †
@…u 1 ; u 2 ; u 3 † du 1du 2 du 3 :
1.23. Evaluate the integral R …1;2†
…0;1† …x2 y†dx ‡…y 2 ‡ x†dy along
(a) a straight line from (0, 1) to (1, 2);
(b) the parabola x ˆ t; y ˆ t 2 ‡ 1;
(c) straight lines from (0, 1) to (1, 1) and then from (1, 1) to (1, 2).
1.24. Evaluate the integral R …1;1†
…0;0† …x2 ‡ y 2 †dx along (see Fig. 1.24):
(a) the straight line y ˆ x,
(b) the circle arc of radius 1 (x 1† 2 ‡ y 2 ˆ 1.
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