Justification for Geometrical Representation of Gradient Operator
Theorem
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 $A$.
Proof
According to the geometrical representation:
Let $S$ be a constant value of $F$ giving rise to an equal surface of $F$.
Let $S + \d S$ be another constant value of $F$ giving rise to an equal surface of $F$ very close to $S$.
To economise on notation we refer to those equal surfaces themselves as $S$ and $S + \d S$ respectively.
Consider the point $A$ on $S$ whose position vector is $\mathbf r$.
Let $B$ be a point on $S + \d S$ whose position vector is $\mathbf r + \d \mathrm r$.
The smallest distance from $S$ to $S + \d S$ from $A$ is $AC$:
- whose direction is that of the unit normal $\mathbf {\hat n}$ at $A$
- whose length is $\norm {\d \mathbf n}$.
Let $\norm {\d \mathrm r}$ be the length of $AB$.
This is the magnitude of the rate of increase at $A$ of $S$ in the direction of $AB$ when $S$ and $S + \d S$ are infinitesimally close together.
The rate of increase is greatest in the direction is that of the unit normal, that is, along $AB$.
Hence at $A$ the rate of increase has value $\dfrac {\partial F} {\partial n}$.
Thus:
- $\dfrac {\partial F} {\partial r} = \dfrac {\partial F} {\partial n} \cos \theta$
where $\theta$ is the angle between $AB$ and $AC$.
Hence if $\mathbf {\hat n}$ is the unit normal to $S$ at any point in a scalar field, the vector quantity $\mathbf {\hat n} \dfrac {\partial F} {\partial n}$ gives the greatest rate of increase of $F$ at that point in magnitude and direction.
This is the gradient of $F$ at that point and can be written:
- $\grad F = \dfrac {\partial F} {\partial n} \mathbf {\hat n}$
Thus the gradient of $F$ is a vector field whose direction is the fastest rate of increase of $F$ and whose magnitude is that rate of increase.
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