Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5.13 EXERCISES

5.9 The functionf(x, y) satisfies the differential equation

y

∂f

∂x

+x

∂f

∂y

=0.

By changing to new variablesu=x^2 −y^2 andv=2xy, show thatfis, in fact, a
function ofx^2 −y^2 only.
5.10 Ifx=eucosθandy=eusinθ, show that

∂^2 φ

∂u^2

+

∂^2 φ

∂θ^2

=(x^2 +y^2 )

(

∂^2 f

∂x^2

+

∂^2 f

∂y^2

)

,

wheref(x, y)=φ(u, θ).
5.11 Find and evaluate the maxima, minima and saddle points of the function

f(x, y)=xy(x^2 +y^2 −1).

5.12 Show that

f(x, y)=x^3 − 12 xy+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 function

f(x, y)=(x^2 − 2 y^2 )exp[−(x^2 +y^2 )/a^2 ],
whereais a non-zero constant.
Sketch the function along thex-andy-axes and hence identify the nature and
values of the stationary points.
5.14 Find the stationary points of the function

f(x, y)=x^3 +xy^2 − 12 x−y^2

and identify their natures.
5.15 Find the stationary values of

f(x, y)=4x^2 +4y^2 +x^4 − 6 x^2 y^2 +y^4

and classify them as maxima, minima or saddle points. Make a rough sketch of
the contours offin the quarter planex, y≥0.
5.16 The temperature of a point (x, y, z) on the unit sphere is given by

T(x, y, z)=1+xy+yz.

By using the method of Lagrange multipliers, find the temperature of the hottest
point on the sphere.
5.17 A rectangular parallelepiped has all eight vertices on the ellipsoid

x^2 +3y^2 +3z^2 =1.

Using the symmetry of the parallelepiped about each of the planesx=0,
y=0,z= 0, write down the surface area of the parallelepiped in terms of
the coordinates of the vertex that lies in the octantx, y, z≥0. Hence find the
maximum value of the surface area of such a parallelepiped.
5.18 Two horizontal corridors, 0≤x≤awithy≥0, and 0≤y≤bwithx≥0, meet
at right angles. Find the lengthLof 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 areaAthroughout
its length. The cross-section is to be a rectangle of wall heighth(fixed) and
widthw, surmounted by an isosceles triangular roof that makes an angleθwith