231 Tutorial Sheet 3: due Friday November 412
Questions
1. Rewrite the integral
21 October 2005
I=
Useful facts:
• Some trigonometric integrals are required. In particular you may quote the integrals:
Z 2π
dθ cos θ = 0,
Z 02π
dθ cos2 θ = π,
Z0 2π
dθ cos3 θ = 0
0
Z 2π
3
dθ cos4 θ =
π
(1)
4
0
(2)
1
dy
0
Z
grad φ =
x = a cosh u cos v
y = a sinh u sin v.
F=
• The Laplacian:
4=
1
2
∂2
∂2
∂2
+
+
∂x2 ∂y 2 ∂z 2
(4)
(5)
(6)
Conor Houghton, houghton@maths.tcd.ie, see also http://www.maths.tcd.ie/~houghton/231
Including material from Chris Ford, to whom many thanks.
1
r̂
r
= 3
r2
r
is irrotational (here r = xi + yj + zk and r = |r| =
• For a vector field F = (F1 , F2 , F3 ) the divergence is
• For a vector field F = (F1 , F2 , F3 ) the curl is
i
∂ ∂j k
∂
curl F = ∇ × F = ∂x
∂y
∂z
F1 F2 F3
(8)
(9)
4. Show that away from the origin the vector field
(3)
∂F1 ∂F2 ∂F3
+
+
∂x
∂y
∂z
(7)
3. Compute the area and centroid of the plane region enclosed by the cardioid r(θ) =
1 + cos θ (r and θ are polar coordinates).
(10)
p
x2 + y 2 + z 2 ).
∇ × (∇ × F) = ∇(∇ · F) − 4F.
∂φ
∂φ
∂φ
i+
j+
k
∂x
∂y
∂z
div F = ∇ · F =
dx φ(x, y),
tan−1 y
2. Compute the element of area for elliptic cylinder coordinates which are defined as
5. Prove the identity
• For a scalar field φ the gradiant is
π
4
as an iterated double integral with the opposite order of integration. Compute the
area of the region of integration.
Two are zero by symmetry, the other two can be computed through standard trigonometric identities or via complex exponentials:
1
cos θ = (eiθ + e−iθ ).
2
Z
2
(11)