Gradient Fields

A gradient field is a vector field derived from a scalar field. A scalar field assigns a single numerical value to every point in space, such as temperature, pressure, or electric potential. The gradient field describes how that value changes from point to point. It gives both the direction of the fastest increase and the rate of change in that direction.

For a scalar field f(x, y), the gradient is written as

\begin{equation*}\nabla f=\left\langle \jfrac{\partial f}{\partial x},\jfrac{\partial f}{\partial y}\right\rangle \end{equation*}

In three dimensions, for f(x, y, z), it is

\begin{equation*}\nabla f=\left\langle \jfrac{\partial f}{\partial x},\jfrac{\partial f}{\partial y},\jfrac{\partial f}{\partial z}\right\rangle \end{equation*}

Each component is a partial derivative, which measures how the scalar field changes along one coordinate direction while the other variables are held constant.

The gradient points in the direction where the scalar field increases most rapidly. Its magnitude,

\begin{equation*}|\nabla \textit{f}|=\bm{\sqrt{\Biggl(\jfrac{\partial \textit{f}}{\partial \textit{x}}\Biggr)^2+\Biggl(\jfrac{\partial \textit{f}}{\partial \textit{y}}\Biggr)^2+\Biggl(\jfrac{\partial \textit{f}}{\partial \textit{z}}\Biggr)^2}}\end{equation*}

represents the maximum rate of increase at that point.

Gradient vectors are perpendicular to level curves in two dimensions and to level surfaces in three dimensions. A level curve is defined by f(x, y) = c, where c is a constant. Along this curve, the scalar value does not change, so the gradient points normal to it.

Where ∇f = 0, the scalar field has no direction of immediate increase. These points may represent local maxima, minima, or saddle points, depending on the field's behavior nearby.