Physics 89 (Mathematical Methods)
Problem Set #3
Due by 6pm, September 28, 2021
1
Area of a triangle
Using the cross-product, calculate the area A of the triangle with vertices
(a, 0, 0); (0, b, 0); (0, 0, c).
Express A in terms of the areas of the three projections of the triangle onto the x̂ − ŷ, x̂ − ẑ and
ŷ − ẑ planes. (These areas are ab/2, ac/2, and bc/2 respectively.)
2
Cross product application
~ B,
~ C
~ in 3D are arranged so that the angle between any pair is θ. Use
(a) Three unit vectors A,
~ × B)
~ · (C
~ × D)
~ = (A
~ · C)(
~ B
~ · D)
~ − (A
~ · D)(
~ B
~ · C)
~ to
the cross-product and the identity (A
~ and B
~ and the plane that contains A
~
calculate the angle between the plane that contains A
~
and C.
(b) As an application, calculate the angle between adjacent faces of the dodecahedron (which is
the polyhedron made of 12 regular pentagons glued along edges with three faces meeting at
.
each vertex). In this case θ = 3π
5
Hint: calculate the angle between the normal vectors to the planes.
~ × B)
~ · (C
~ × D)
~ = (A
~ · C)(
~ B
~ · D)
~ − (A
~ · D)(
~ B
~ · C)
~ using the Levi-Civita
(c) Prove the identity (A
and Kronecker symbols and the identity
ijk lmk = δil δjm − δim δjl
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
3
Contour integrals
y
C
z = x + iy
C
−R
x
R
Calculate
Z
∞
A=
−∞
dx
4
x +1
Z
and
∞
B=
−∞
cos x dx
x4 + 1
by following the steps below:
(a) Find the 4 complex solutions to z 4 + 1 = 0.
These would be the 4 simple poles of the analytic functions relevant to our problem.
You can express the answer in polar form.
(b) Find the residues of the function f (z) = z41+1 and g(z) =
part (a).
You can express the answer in polar form.
eiz
z 4 +1
at the four simple poles from
(c) Use a contour C that extends along the real axis from −R to R and then back
R dxto −R along a
) as R → ∞.
semicircle Hin the upper half
plane
(the
same
contour
we
studied
in
class
for
x2 +1
H
Calculate f (z)dz and g(z)dz.
Remember to write the results as real numbers.
(d) Explain in up to three sentences why the answers to part (c) gives A and B.