Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 6
Due: at the end of the tutorial session Tuesday/Thursday, 8/10 March 2016
Name and student number:
1.
(a) Express rectangular coordinates in terms of spherical coordinates.
Draw the corresponding picture.
Compute the volume of a ball of radius R.
(b) Consider the solid G bounded by the surfaces x2 + y 2 + z 2 = 4 and x2 + y 2 + z 2 = 16
and below by the surface z = 0.
1. What is the surface x2 + y 2 + z 2 = 4?
2. What is the surface x2 + y 2 + z 2 = 16?
3. What is the surface z = 0?
4. Sketch the part of the boundary of the solid G which belongs to the surface z = 0.
5. Use triple integral and spherical coordinates to compute the volume V of the solid G.
6. Use triple integral and spherical coordinates to find the mass M of the solid G if its
density is
1
2
2
2
e− 4 (x +y +z )
.
δ(x, y, z) = p
x2 + y 2 + z 2
Show the details of your work.
2.
(a) Express rectangular coordinates in terms of cylindrical coordinates
(b) Consider the solid G bounded by the surfaces x2 + y 2 = 4 and x2 + y 2 = 9, above by
the surface z = 13 − x2 − y 2 , and below by the surface z = 0.
i. What are the surfaces x2 + y 2 = 4 and x2 + y 2 = 9?
ii. What is the surface z = 13 − x2 − y 2 ?
iii. What is the surface z = 0? Sketch the part of the boundary of the solid G which belongs
to the surface z = 0.
iv. Use triple integral to compute the volume V of the solid G.
v. Use triple integral to find the mass M of the solid G if its density is
δ(x, y, z) = e12−x
1
2 −y 2 −z
.
Show the details of your work.
3. Let r = xi + yj + zk, let r = ||r||, let f be a differentiable function of one variable, and
let F(r) = f (r)r. Show that
(a) curl r = 0 ,
(a0 ) curl F = 0 ,
(b) ∇ r =
r
,
r
(b0 ) ∇ f (r) =
(c) div r = 3 ,
f 0 (r)
r,
r
2
(d) ∇
1
r
=− 3,
r
r
(c0 ) div F = 3f (r) + rf 0 (r) .