# Thread: Gradient and Directional Derivative

1. ## 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.

2. Well, $\displaystyle \nabla A(x,y) = -2x \hat{i} - 2y \hat{j}$

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 $\displaystyle -\nabla A(x,y)$

at the point (3,4) , $\displaystyle -\nabla A(x,y) = 6 \hat{i} + 8 \hat{j}$

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

3. Of course, the vector $\displaystyle 6\vec{i}+ 8\vec{j}$ has angle $\displaystyle tan^{-1}\frac{4}{3}$ with the positive x axis, NOT $\displaystyle tan^{-1} 10$.

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 $\displaystyle 3\vec{i}+ 4\vec{j}$ which is in the same direction as $\displaystyle 6\vec{i}+ 8\vec{j}$.

4. Originally Posted by HallsofIvy
Of course, the vector $\displaystyle 6\vec{i}+ 8\vec{j}$ has angle $\displaystyle tan^{-1}\frac{4}{3}$ with the positive x axis, NOT $\displaystyle tan^{-1} 10$.

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 $\displaystyle 3\vec{i}+ 4\vec{j}$ which is in the same direction as $\displaystyle 6\vec{i}+ 8\vec{j}$.
So 6i + 8j would be the direction of the runoff? Or could I also say tan-1(4/3)=53.1degrees is the direction.