Directional Derivatives
By Zainab Zanoba on Thursday, November 15, 2001 - 10:38 pm:

How can I solve this problem:
a metal plate is locatedin an xy-plane such that the temperature T at (x,y) is inversely proportional to the distance from the origin , and the temprature at p(3,4) is 100F.

Find the rate of change of T at P in the direction of i+j

By Dan Goodman on Thursday, November 22, 2001 - 01:46 pm:
Two steps, first we formulate the problem: we know that
T(x,y)=A/   _____
Öx2+y2
 

because
  _____
Öx2+y2
 

is the distance from the origin, and A is some constant. If we know T(3,4)=100 then we have that
100=A/   _____
Ö32+42
 

which we can solve for A.

You might not know the formula for the rate of change of a function in two variables in a particular direction, what you do is if n=(u,v) is the direction you are interested in then the rate of change of T in the direction n is

ÑnT=ÑT.n/|n|

=(T/x, T/y).(u,v)/|(n|

=(u/|n|)T/x+(v/|n|)T/y

In your example we have n=(u,v)=(1,1) so u=v=1 and n = Ö2. Can you work out T/x and T/ y? If so, just substitute x=3, y=4 into (1/Ö2)( T/x+T/y)(3,4).