Gradient vs. Level Curves

Move the yellow point to see how the gradient of the function (whose graph is the red surface) changes. Note that (1) the gradient is always perpendicular to the level curves, (2) though the gradient points in the direction of greatest change, this direction is not always directly toward the "summit" (i.e. the local maximum), and (3) the magnitude (length) of the gradient is equal to the slope of the tangent line to the surface in the direction of the gradient (this is the solid black line on the right side).