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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?

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.

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