Thread: Gradient and Directional Derivative

A geological map shows that the altitude at point (x,y) is
A(x,y)=100-x^2-y^2 feet. If water is spilled at (3,4), in which direction will it run off?

I used the angle of inclination and ended up with tan^-1(10) = 84 degrees but Im not sure if this is the correct method.

Well,

The gradient is the direction of the greatest increase. But I think what we want is the direction of the greatest decrease, which is simply

at the point (3,4) ,

I would just leave the answer like that (or normalize it if you like).

Of course, the vector has angle with the positive x axis, NOT .

I might also point out that the contour lines here are circles so lines perpendicular to the circle are radii. The radius through (3, 4) has direction vector which is in the same direction as .

Originally Posted by HallsofIvy

Of course, the vector has angle with the positive x axis, NOT .

I might also point out that the contour lines here are circles so lines perpendicular to the circle are radii. The radius through (3, 4) has direction vector which is in the same direction as .

So 6i + 8j would be the direction of the runoff? Or could I also say tan-1(4/3)=53.1degrees is the direction.