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.