Mathematical Methods for Physics and Engineering - Matematica.NET

5.13 EXERCISES5.9 The function f(x, y) satisfies the differential equationy ∂f∂x + x ∂f∂y =0.By changing to new variables u = x 2 − y 2 and v =2xy, show that f is, in fact, afunction of x 2 − y 2 only.5.10 If x = e u cos θ and y = e u sin θ, show that( )∂ 2 φ∂u + ∂2 φ∂ 2 ∂θ 2 =(x2 + y 2 2 f)∂x + ∂2 f,2 ∂y 2where f(x, y) =φ(u, θ).5.11 Find and evaluate the maxima, minima and saddle points of the functionf(x, y) =xy(x 2 + y 2 − 1).5.12 Show thatf(x, y) =x 3 − 12xy +48x + by 2 , b ≠0,has two, one, or zero stationary points, according to whether |b| is less than,equal to, or greater than 3.5.13 Locate the stationary points of the functionf(x, y) =(x 2 − 2y 2 )exp[−(x 2 + y 2 )/a 2 ],where a is a non-zero constant.Sketch the function along the x- andy-axes and hence identify the nature andvalues of the stationary points.5.14 Find the stationary points of the functionf(x, y) =x 3 + xy 2 − 12x − y 2and identify their natures.5.15 Find the stationary values off(x, y) =4x 2 +4y 2 + x 4 − 6x 2 y 2 + y 4and classify them as maxima, minima or saddle points. Make a rough sketch ofthe contours of f in the quarter plane x, y ≥ 0.5.16 The temperature of a point (x, y, z) on the unit sphere is given byT (x, y, z) =1+xy + yz.By using the method of Lagrange multipliers, find the temperature of the hottestpoint on the sphere.5.17 A rectangular parallelepiped has all eight vertices on the ellipsoidx 2 +3y 2 +3z 2 =1.Using the symmetry of the parallelepiped about each of the planes x = 0,y =0,z = 0, write down the surface area of the parallelepiped in terms ofthe coordinates of the vertex that lies in the octant x, y, z ≥ 0. Hence find themaximum value of the surface area of such a parallelepiped.5.18 Two horizontal corridors, 0 ≤ x ≤ a with y ≥ 0, and 0 ≤ y ≤ b with x ≥ 0, meetat right angles. Find the length L of the longest ladder (considered as a stick)that may be carried horizontally around the corner.5.19 A barn is to be constructed with a uniform cross-sectional area A throughoutits length. The cross-section is to be a rectangle of wall height h (fixed) andwidth w, surmounted by an isosceles triangular roof that makes an angle θ with181