# Gradients

A metal plate is located in an xy-plane such that the temperature T at (x,y) is inversely proportional to the distance from the origin, and the temperature at point P(3,4) is 100 (i.e. the temperature at any point (x,y) is described by the function

T(x,y) = 500/(x^2 + y^2)^1/2

a) in what direction does the T increase most rapidly at P? Write the vector representing that direction explicitly.

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

c) In what direction does T decrease most rapidly at P?

Â© BrainMass Inc. brainmass.com February 24, 2021, 2:16 pm ad1c9bdddfhttps://brainmass.com/math/calculus-and-analysis/gradient-plane-functions-10493

#### Solution Preview

a.)

The gradient of T at (3,4) will give the maximum rate of change of T at (3,4)

grad(T(x,y)) = (del(T)/del(x)).i + (del(T)/del(y)).j

where, del(T)/del(x) is the partial derivative of T with respect to x.

because,

T = 500/(x^2 + y^2)^1/2

hence,

del(T)/del(x) = 500x/(x^2 + ...

#### Solution Summary

This shows how to determine direction of increase (and the vector), rate of change, and direction of decrease.