Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
Problems Class 8: Solutions
1. Solve the following dierential equations:
2y
= 9 y,
x2
with boundary condition y(0) = 5 and y (0)=0. The easiest way is to recognize the
two solutions, y1 (x) = A sin 3x and y2 (x) = B cos 3x,
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
(J A Dunningham)
Assignment 1: Solutions. Number of points given in [brackets]. 29 points maximum.
1. The volume can be calculated using the scalar triple product, using the three vectors
emanating from any vertex of the pa
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
(J A Dunningham)
Assignment 2: Solutions. Number of points given in [brackets]. 30 points maximum.
1. We compute the integral over the square, and subtract the integral over the circle. [5]
3
3
1
dy 2x2 = 2 2 33 (6) = 216
3
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
Problems Class 3: Solutions
1. (a)
sin(q r) = cos(q r) (q r) = q cos(q r) (using the fact that
(q r) = q).
(b)
r = 0 (You need coordinates for this; just calculate the curl in the usual way
using r = x i + y j + z k). Alte
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
Problems Class 2: Solutions
1. The equation of motion is mv = q [v B + E], where q = e and E = E0 x.
(a) At time t = 0 the particle is at rest. Hence
it is rst accelerated in the direction antiparallel to the electric eld E
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
Problems Class 11: Solutions
The Fourier Series for a function f (x) with period L is
f (x) =
an =
2
L
2nx
L
a0
+
an cos
2
n=1
L/2
f (x) cos
L/2
+
2nx
L
bn sin
n=1
2nx
L
dx,
bn =
2
L
L/2
f (x) sin
L/2
2nx
L
dx.
1. (a) S(x)
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
Problems Class 6: Solutions
1. (a) The area element is dS = d dz. The cylindrical surface is dened by =
R, : 0 2, z : 0 H. The unit vector is a function of angle according to
= cos i + sin j. So,
2
I1 =
dS = R
H
d
S
dz cos
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
Problems Class 10
1. An elastic membrane is held around its edge by a rectangular frame that has a side of
length a in the xdirection and of length b in the ydirection. The frame also holds the
height u(x, y) to be zero o
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
Problems Class 7: Solutions
1. (a) We can identify
V=
y
1
sin i cos j
i j =
=
2
2
x
x
r cos
r cos2
(b) To show its conservative, take the curl:
V =
1
1
2 = 0.
x2 x
(1)
The curl vanishes, so V is conservative.
(c) To const
Vector Calculus and Partial Differential Equations
PHYS 2170

Fall 2013
PHYS2170 Mathematical Methods 4
Problems Class 9
1. Show explicitly that the following functions satisfy the partial dierential equation
c f = f :
x
t
(a) f (x, t) = sinh(x + ct)
(b) f (x, t) = g(x + ct), where g is any dierentiable function.
f (x, t) = s