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Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction <math>u</math> is maximal in the direction of the gradient, and equal to <math>||\nabla f||</math> for <math>u</math> a unit vector in the direction of the gradient.
Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction <math>u</math> is maximal in the direction of the gradient, and equal to <math>||\nabla f||</math> for <math>u</math> a unit vector in the direction of the gradient.
==Properties of the Gradient==
If <math>f</math> is a differentiable function with smooth level sets <math>f^{-1}(c)</math>, then the gradient vector field <math>\nabla f</math> is perpendicular to the level sets of <math>f</math>. For fix a level set <math>S = f^{-1}(c)</math>, and let <math>v</math> be a vector tangent to <math>S</math> at <math>p</math>. Then we can find a curve <math>\gamma(t)</math> on <math>S</math> with <math>\gamma'(0) = v</math>. Now
<math>
f\circ\gamma(t) = c
</math>
since <math>S</math> is a level set. Taking derivatives of both sides and applying the chain rule, we get that
<math>
\nabla f\cdot \gamma'(0) = \nabla f\cdot v = 0
</math>
Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.
[[Category:mathematics]]
[[Category:mathematics]]