Understanding Engineering Mathematics

science and engineering. An important example is Laplace’s equation

∂^2 f

∂x^2

+

∂^2 f

∂y^2

= 0

which arises in electromagnetic field theory and fluid flow, for example. Verify that

each of the following functions satisfies Laplace’s equation

(i) f(x,y)=ln(x^2 +y^2 ) (ii) f(x,y)=e−^3 ycos 3x

(iii) f(x,y)=ex(xcosy−ysiny) (iv) f(x,y)=x^2 −y^2

2.An important application of the total differential occurs in estimating the change in a
function of a number of variables resulting from changes in the variables. The following
give a number of examples of this.
(i) From the ideal gas lawPV=nRT,wherenRis constant, estimate the percentage
change in pressure,P, if the temperature,T, is increased by 3% and the volume,
V, is increased by 4%.
(ii) Given that time,T, of oscillation of a simple pendulum of lengthlis

T= 2 π

√

l

g

determine the total differentialdTin terms ofland the gravitational and constant

g. Estimate the percentage error in the period of oscillation iflis taken 0.1% too

large andg0.05% too small.

(iii) The total resistance of three resistorsR 1 ,R 2 ,R 3 , in parallel is given byR,where

1

R

=

1

R 1

+

1

R 2

+

1

R 3

IfR 1 ,R 2 ,R 3 , are measured as 6, 8, 12respectively with respective maximum

tolerances of±0.1,±0.03,±0.15, estimate the maximum possible error inR.

3.The total derivative is used to determine the rate of change of a function of a number
of variables in terms of the rates of change of the variables. The following examples
illustrate this.
(i) The radius of a cylinder decreases at a rate 0.02 ms−^1 while its height increases
at a rate 0.01 ms−^1. Find the rate of change of the volume at the instant when
r= 0 .05 m andh= 0 .2m.
(ii) Find the rate of increase of the diagonal of a rectangular solid with sides 3, 4, 5 m,
if the sides increase at^13 ,^14 ,^15 ms−^1 respectively.

16.8 Answers to reinforcement exercises

1.(i) 16 (ii)

π

2

(iii) 3,

√

29 (iv) 8