gradient

Stand on a hillside and ask: which way is steepest uphill, and how steep is it? The gradient answers both at once. It is a single arrow that points in the direction of fastest increase of the function, and whose length is the rate of increase in that best direction. Every other direction gives a smaller climb, governed by how much it aligns with the gradient.

For a scalar function f: R^n -> R, the gradient at a is the vector grad f(a) = (df/dx_1, ..., df/dx_n) of its partial derivatives. When f is differentiable, the directional derivative in a unit direction u equals the dot product grad f(a) . u. By Cauchy–Schwarz this dot product is largest when u points along grad f(a), confirming the steepest-ascent interpretation.

The gradient is perpendicular to level sets: at a point where grad f is nonzero, it is normal to the level surface through that point, because moving along a level set keeps f constant, giving zero directional derivative there. A caveat: the gradient is genuinely a vector only after a choice of inner product; the coordinate-free object is the total derivative (a linear functional), and the gradient is its representative under the dot product.