Understanding Engineering Mathematics

2. Show thatf(x,y)=ln(x^2 +y^2 )satisfies thepartial differential equation:

∂^2 f

∂x^2

+

∂^2 f

∂y^2

= 0

This is called theLaplace equationin two-dimensional rectangular coordinates. It is

very important in fluid mechanics, electromagnetism, and many other areas of science

and engineering, as well as being a key equation in pure mathematics.

Answers

1. (i)fx= 3 x^2 y^2 + 4 y^4 ,fy= 2 x^3 y+ 16 xy^3 ,

fxx= 6 xy^2 ,fyy= 2 x^3 + 48 xy^2 ,fxy= 6 x^2 y+ 16 y^3

(ii) fx=exy(ycos(x+y)−sin(x+y)),

fy=exy(xcos(x+y)−sin(x+y)),

fxx=exy((y^2 − 1 )cos(x+y)− 2 ysin(x+y)),

fyy=exy((x^2 − 1 )cos(x+y)− 2 xsin(x+y)),

fxy=exy(xycos(x+y)−(x+y)sin(x+y))

16.5 The total differential

∂z

∂x

,

∂z

∂y

give the rates of increase ofz=f(x,y)in thexandy directions respec-

tively – what about the increase in a general direction? I.e. ifx,yincrease byδx,δy,

by how much doeszincrease? From Figure 16.6 we can see that the increase inz,δz,is

given approximately by

δz

δz

δx

δx+

δz

δy

δy

Such diagrams are not to everyone’s taste, so a few examples might help.

Problem 16.3

Obtain expressions fordzin the cases

(i) z=xYy (ii) z=xy (iii) z=x^2 Yy^2

What happens ifdx,dyare so small that products of them can be neglected?

(i) is not so bad:

δz=x+δx+y+δy−(x+y)=δx+δy

This result is exact. There are no products ofδx,δy.

(ii) is a little more complicated:

δz=(x+δx)(y+δy)−xy

=yδx+xδy+δxδy