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).
I found the partial derivatives(easy lol)
Tx = 2x
Ty = 3y
Thank you for help