[SOLVED] Physics 311 Homework Set 3

1. Check the fundamental theorem for gradients using T(x, y, z)=3z2 +4xz +yz3, the points
~a = (0, 0, 0) , ~b = (1, 1, 1) , and the three paths in Fig. 1.28:
a) (0, 0, 0) ! (1, 0, 0) ! (1, 1, 0) ! (1, 1, 1);
b) (0, 0, 0) ! (0, 0, 1) ! (0, 1, 1) ! (1, 1, 1);
c) the parabolic path z = x2 ; y = x.2. Test the Divergence theorem for the function
~v = (y2
z2
)x + (3z2
x2
)y + (42
y2
)z. (1)
Take as your volume the cube shown in Fig. 1.30, with sides of length 2.3. Test Stokes theorem for the function
~v = (y2
z2
)x + (3z2
x2
)y + (42
y2
)z. (2)
using the triangular shaded area of Fig. 1.34.4. Compute the divergence of the function
~v = (r2 sin )r + (r2 cos ) + (r2 sin cos ). (3)
Check the Divergence theorem for this function, using as your volume the inverted
hemispherical bowl of radius R, resting on the xy plane and centered at the origin
(see Fig. 1.40).5. a) Find the divergence of the function
~v = s2
(3 + sin2 )s + s2 sin cos + 5z2
z (4)b) Test the divergence theorem for this function, using the quarter-cylinder
(radius 2, height 5), shown in Fig. 1.43.
c) Find the curl of ~v.6. Let
F~1 = y2
x + z2
y + x2
z
F~2 = x2
x + y2
y + z2
z (5)a) Calculate the divergence and the curl of F~1 and F~2. Which one can be written as the
gradient of a scalar? Find a scalar potential that does the job. Which one can be
written as the curl of a vector? Find a suitable vector potential.b) Show that F~3 = (2yz)x + (2zx)y + (2xy)z can be written both as the gradient of a
scalar and as the curl of a vector. Find scalar and vector potentials for this function.