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 / \sqrt{x^{2} + y^{2}}

because

\sqrt{x^{2} + y^{2}}

is the distance from the origin, and

A

is some constant. If we know

T (3, 4) = 100

then we have that

100 = A / \sqrt{3^{2} + 4^{2}}

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

$\nabla_{n} T = \nabla T . n / | n |$

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

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

In your example we have $n = (u, v) = (1, 1)$ so $u = v = 1$ and $n = \sqrt{2}$ . Can you work out $\partial T / \partial x$ and $\partial T / \partial y$ ? If so, just substitute $x = 3$ , $y = 4$ into $(1 / \sqrt{2}) (\partial T / \partial x + \partial T / \partial y) (3, 4)$ .