A scalar field involves numerical values only, without direction. Then why does the gradient of it become a vector field?
Thanks!
$\endgroup$3 Answers
$\begingroup$Because the gradient tells you about directional derivatives.
$\endgroup$ $\begingroup$Think of the height of a point on a hill. The height is a scalar, but the gradient of the height gives both direction and magnitude of the slope at the point.
$\endgroup$ $\begingroup$The directions are not in the values of the function $f$ you are studying but in the ways you can move around in space. Standing at a point $p_0$in the domain of $f$ you can move away along rays in all possible directions. It is a fact of life that in most cases the initial rate of change of $f$ along such a ray depends on the chosen direction. When the function $f$ is differentiable at $p_0$ it does so in a characteristic way, which can be encoded in a vector, the gradient $\nabla f(p_0)$ of $f$ at $p_0$, and is exhibited in the following miraculous formula: $$f(p_0+X)-f(p_0)=\nabla f(p_0)\cdot X+o(|X|)\qquad(X\to 0)\ .$$
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