# Moving particle problem

• May 12th 2012, 02:33 AM
Brennox
Moving particle problem
So i dont really know how to go about this question, if u can please show the method leading up to the solution

The temperature at each point (x,y) of a metal plate is given by

T(x,y)= 1 + x^2 + 3/2(y^2)

The path of the heat-seeking particle on the plate is a curve r(t) = (x(t), y(t)) such that for each t, the velocity r'(t) of the particle at time t is in the direction of the fastest increase of the temperature T at r(t). Suppose that particle starts at the point (1,1) wih speed |r'(0)|=8 units.

a) Determine the equation for r'(t) thus obtaining differential equations for the functions x(t) and y(t). What initial conditions do these functions satisfy at t=0?

b) Use part (a) to ﬁnd explicitly the functions x(t) and y(t) and thus the path r(t).

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I found the partial derivatives(easy lol)

Tx = 2x

Ty = 3y

Thank you for help
• May 17th 2012, 02:27 PM
TD!
Re: Moving particle problem
The direction of steepest increase of a function f is given by its gradient, grad(f) or ∇f; which you already calculated to be (2x,3y). You don't know the exact value of the speed r'(t) but you do know its direction (the gradient), so both are multiples of eachother:

$(\vec r)'(t) = k\nabla f \Leftrightarrow (x'(t),y'(t)) = (2kx,3ky)$

This is an easy system of two linear differential equations that you can solve combined with the initial values x(0) = y(0) = 1. The given initial speed will allow you to determine the constant of proportionality k that we introduced. Does that help?