Definition:Gradient Operator/Cartesian 3-Space

Definition

Let $R$ be a region of Cartesian $3$-space $\R^3$.

Let $\map F {x, y, z}$ be a scalar field acting over $R$.

Let $\tuple {i, j, k}$ be the standard ordered basis on $\R^3$.

The gradient of $F$ is defined as:

\(\ds \grad F\)

\(:=\)

\(\ds \nabla F\)

\(\ds \)

\(=\)

\(\ds \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } F\)

Definition of Del Operator

\(\ds \)

\(=\)

\(\ds \dfrac {\partial F} {\partial x} \mathbf i + \dfrac {\partial F} {\partial y} \mathbf j + \dfrac {\partial F} {\partial z} \mathbf k\)

Also see

• 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: $2 a$. The Operation $\nabla S$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: The Gradient: $22.29$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): gradient: 2. (grad)
• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 13.6$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): gradient: 2. (grad)

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