Thread: Level curves and direction of greatest increase

Question

The temperature T in a metal disc in the x-y plane is given by T=sqrt(x^2+y^2).

Explain why, at any point in the disc, the direction of greatest increase in temperature is given by a vector that points away from the origin.

Attempt

The direction of greatest in crease is givne by |grad(T)|. The level curves are circles centred at the origin with the equation x^2+y^2= k (k=constant).

From here I am totally unsure what to do...

The gradient is normal to the level curves. If you know that the level curves are concentric circles centred at the origin, then you know that the gradient points either toward the origin or away from the origin, because these are the only directions normal to circles centred at the origin. Its easy to see that the function increases as you move away from the origin, so the gradient must point away from the origin.

Alternatively, you can just calculate the gradient at a point p and see that it is proportional to p.