Answer
Two main geometric properties:
1. $\nabla {f_P}$ points in the direction of maximum rate of increase. The maximum rate of increase is $||\nabla {f_P}||$.
2. $\nabla {f_P}$ is orthogonal to the level curve (or surface) through $P$.
Work Step by Step
Assume $\nabla {f_P} \ne {\bf{0}}$. Two main geometric properties of the gradient $\nabla f$ at the point $P$:
1. $\nabla {f_P}$ points in the direction of maximum rate of increase. The maximum rate of increase is $||\nabla {f_P}||$.
2. $\nabla {f_P}$ is orthogonal to the level curve (or surface) through $P$.
