Definition:Gradient Operator/Geometrical Representation

Definition

Let $R$ be a region of space.

Let $F$ be a scalar field acting over $R$.

The gradient of $F$ at a point $A$ in $R$ is defined as:

$\grad F = \dfrac {\partial F} {\partial n} \mathbf {\hat n}$

where:

$\mathbf {\hat n}$ denotes the unit normal to the equal surface $S$ of $F$ at $A$

$n$ is the magnitude of the normal vector to $S$ at $A$.

Also see

• Justification for Geometrical Representation of Gradient Operator

• Results about the gradient operator can be found here.

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $2$. The Gradient of a Scalar Field: $(4.2)$