Basic Question on Gradients

Question

Hi I am having trouble understanding how the gradient of a scalar field is the direction along the xy plane that yields the maximum inclination. Sure it takes into account the partial derivatives along both the x and y axes but does that mean "maximum inclination"? How do the partial derivatives come into the gradient vector field?
A basic intuition would be great.
Sorry for phrasing the question badly...I am a high school student venturing on some multivariate calculus so please bear with me.

Thank You in advance.

Bidit Acharya
Kathmandu, Nepal

Answer 1

For a smooth function $f: \mathbb{R}^2 \to \mathbb{R}$, the gradient at a point $(x_0, y_0)$ is perpendicular to the level curve at that point. If $c = f(x_0, y_0)$, the level curve is $f(x,y) = c$. If you move along a level curve, then the value of $f$ does not change at all. If you move perpendicular to the level curve (i.e., in the direction of the gradient), then you are moving "as fast as possible" away from the level curve. That is what is meant by "maximum inclination"; if you think of $f$ as the height of a mountain, then the gradient is the direction which is locally the steepest.

The word locally is important. The gradient does not usually point to the top of the mountain. Once you leave the original point, the gradient may change direction and thus the locally steepest direction will change.

Answer 2

First note that for a unit vector $v\in \mathbb{R}^n$ and a function which is differentiable in $x\in \mathbb{R}^n$, the derivative in the direction $v$ (which is defined as the limit of $(f(x+vt)-f(x))/t$ as $t\to 0$) is $\nabla f(x) \cdot v$ (use the chain rule to verify this).

Assuming $\nabla f(x)\ne 0$, the question is for which unit vector $v$ the number $\nabla f(x) \cdot v$ is maximal, and it turns out that this is the case for $v=\nabla f(x)/|\nabla f(x)|$ (the direction of the gradient); in fact this is precisely the Cauchy-Schwarz inequality.