MATH0019 Homework 2 Due: October 28, 2022 Basic 1. Consider polar coordinates in the ry-plane given by the equations r = vx2 +y?, 0 = tan-'(y/x), which can be solved for r, y to give I = roos0, y = rsin0. (a) Find the partial derivatives ar dy ay' dy ar ' (b) If f(x, y) is a differentiable function of r and y, find Of /Or and Of /20. (c) Show that + (1) Remark: The left-hand side of (1) is the norm of the gradient vector Vf squared. This equation gives the norm of Vf in polar coordinates in R?. (d) Show that the Laplace operator A= + dy 2 is given in polar coordinates by 02 10 10 r or 12 202 i.e. for a function f(z, y) which has continuous second partial derivatives we have of of of 10f 102f 12. Let f and g be two differentiable functions with continuous second partial derivatives on R. Fix a constant a. Let u=x-at, v=r+at and define the function w : R' -> R by w(x,t) = f(u) + g(v). Show that w satisfies the wave equation 20-w at2 3. Deduce the Inverse Function Theorem from the Implicit Function Theorem. 4. (a) If f : R -+ R satisfies f'(a) 0 for all a E R, show that f is injective on all of R. (b) (i) Define f : R2 - R' by f(x, y) = (et cosy, e sin y). Show that detf(x,y) # 0 for all (x, y) but f is not injective. (ii) What is the range of f? (ii) Put a = (0, #/2), b = f(a), let g be the continuous inverse of f, defined in an neighbourhood of b, such that g(b) = a. Find an explicit formula for g, compute f'(a) and g'(b) and verify the formula in the inverse function theorem. (iv) What are the images under f of lines parallel to the coordinate axes? 5. Consider the tranformation - y 1 = 3c + 2y Show that the Jacobian determinant is zero on the y-axis. Compute the partial derivatives Or Or dy dy du' du' du' av However, show that the transformation is invertible over the entire ry-plane. To be handed in 1. *Let f and g be two differentiable functions with continuous second partial derivatives on R. Fix a positive constant a. Define the function w : R' - R by w(x,t) = f(x - at) + f(x + at) 2 za Joe gu) du. 2Show that w satisfies the wave equation 20-w =1 and the initial conditions w(x, 0) = f(x), Ow Ty (I, 0) = 9(2). 2. *Consider the pair of equations = R-, 1 close to the point (1,0, VR- - 1), where R is a constant with R > 1. Show that these equations determine the functions r and z implicitly as differentiable functions of y. Find day' ay What happens when R = 1? 3. *Let Mi = ((x,x,p) ER, (x] - 1)2 + (23)2 + (23)? =1}. Show that M, is a 2-dimensional manifold in R3. Let My= {(x, or3) ER', (x] -1)'+(x?)?+(23)? =1, and (x])+(2)?+(23)? = 1). Show that M2 is a 1-dimensional manifold in R3. Describe the curve geometrically. 4. *Let f be a continuously differentiable function defined on R with If(2)| R be a function defined an open set U C R" containing Lo, the interior of L and continuous on L. Moreover, f is assumed to be differentiable on L". Then the following mean-value theorem holds: There exist a point $ 6 L' with f(b) - f(a) = Df(E)(b - a)