Gradient Operator is Invariant under Coordinate Transformation

Theorem

Let $R$ be a region of space in which there exists an scalar field $F$.

Let $\mathbf V$ denote the gradient of $F$.

Then $\mathbf V$ is invariant under a change of coordinate system on $R$.

Proof

The result follows directly from the geometrical representation of the gradient operator.

$\blacksquare$

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