1. ## Directional Derivatives and Gradient

Suppose that the temperature at a point (
x, y) in the plane is given by T(x, y) = xye^(x^2y^2) .

Explain how the quantity
−gradient of T(x, y) is related to the flow of heat in the plane.

Evaluate the integral from 0 to 1 of (gradient of T(x, y)) dot <x, 1> dx and explain what it represents physically.

2. $\displaystyle - \nabla T(x_{0}, y_{0})$ is the direction at the point $\displaystyle (x_{0}, y_{0})$ in which the temperature is decreasing the most rapidly.

And a line integral in a gradient vector field is independent of the path taken. All that matters is the starting point and ending point.

3. Hi there! Thanks for your help but I am still a bit confused over what the integral represents physically.

I calculated it to be (1 - e)/(2e^2) which is less than 0 obviously, but I am not sure what that means.

Thanks again,

FatherMike

4. Since the line integral is path independent, it just represents the difference in temperature between the two points.