Partial Derivatives, find rate of change

Question

I have done part a) of this question. I am confused about part b), as it doesn't say determine rate of change of temperature with respect to anything, so I am confused. Would it be ∂T/∂x + ∂T/dy ?

Answer 1

You are missing the crucial piece of information: directional derivatives. Gradient tells you the direction of steepest increase, but if you move in some other direction, the change is less. If you move in some direction (let's you move with velocity vector $v$ when time $t$ changes), you can write down the chain rule: $$\frac{dT}{dt}=\frac{\partial T}{\partial x}\frac{dx}{dt}+\frac{\partial T}{\partial y}\frac{dy}{dt}$$ Notice that $dx/dt$ and $dy/dt$ are simply the components of velocity vector. You can rewrite this as:

$$\frac{dT}{dt}=\vec{v}\cdot \nabla T$$ So, dot product of velocity (so: direction in which you are looking for rate of change) with the gradient tells you the rate of change. In this case, you probably want a unit length $v$.