MATH 402 - Assignment #2
Due on Monday February 7, 2011
Name —————————————–
Student number —————————
1
Problem 1:
We say that the function f (x1 , . . . , xn ) is positive-homogenous of degree
k in x1 , . . . , xn if
f (λx1 , . . . , λxn ) = λk f (x1 , . . . , xn )
for every λ > 0. Prove that if f (x1 , . . . , xn ) is continuously differentiable and positive-homogenous of degree k, then
n
X
∂f
xi = kf.
∂xi
i=1
2
Problem 2:
Derive the differential equation which must be satisfied by the function
which extremizes the integral
Z b
f (x, y, y 0 , y 00 ) dx
I=
a
with respect to twice-differentiable functions y = y(x) for which
Z b
J=
g(x, y, y 0 , y 00 ) dx
a
possesses a given prescribed value, and with y and y 0 both prescribed
at the end points a and b.
(a) Show that leaving y unspecified at either end point leads to the
condition
∂f ∗
d ∂f ∗
−
(
)=0
∂y 0
dx ∂y 00
at that end point, where f ∗ = f + λg.
(b) Show that leaving y 0 unspecified at either end point leads to the
condition
∂f ∗
=0
∂y 00
at that end point.
3
Problem 3:
Among all curves joining a given point (0, b) on the y-axis to a point on
the x-axis and enclosing a fixed given area S together with the x-axis,
find the curve which generates the least area when rotated about the
x-axis.
4
Problem 4:
Find the geodesics on the sphere of radius r using the method of Lagrange multipliers applied to a variational problem with a finite equation constraint.