Product Rule for Divergence
Theorem
Let $\map {\mathbf V} {x_1, x_2, \ldots, x_n}$ be a vector space of $n$ dimensions.
Let $\mathbf A$ be a vector field over $\mathbf V$.
Let $U$ be a scalar field over $\mathbf V$.
Then:
- $\map {\operatorname {div} } {U \mathbf A} = \map U {\operatorname {div} \mathbf A} + \mathbf A \cdot \grad U$
where
- $\operatorname {div}$ denotes the divergence operator
- $\grad$ denotes the gradient operator
- $\cdot$ denotes dot product.
Proof
From Divergence Operator on Vector Space is Dot Product of Del Operator and definition of the gradient operator:
\(\ds \operatorname {div} \mathbf V\) | \(=\) | \(\ds \nabla \cdot \mathbf V\) | ||||||||||||
\(\ds \grad \mathbf U\) | \(=\) | \(\ds \nabla U\) |
where $\nabla$ denotes the del operator.
Hence we are to demonstrate that:
- $\nabla \cdot \paren {U \, \mathbf A} = \map U {\nabla \cdot \mathbf A} + \paren {\nabla U} \cdot \mathbf A$
Let $\mathbf A$ be expressed as a vector-valued function on $\mathbf V$:
- $\mathbf A := \tuple {\map {A_1} {\mathbf r}, \map {A_2} {\mathbf r}, \ldots, \map {A_n} {\mathbf r} }$
where $\mathbf r = \tuple {x_1, x_2, \ldots, x_n}$ is an arbitrary element of $\mathbf V$.
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.
Then:
\(\ds \nabla \cdot \paren {U \mathbf A}\) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \frac {\map \partial {U A_k} } {\partial x_k}\) | Definition of Divergence Operator | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \paren {U \frac {\partial A_k} {\partial x_k} + \frac {\partial U} {\partial x_k} A_k}\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds U \sum_{k \mathop = 1}^n \frac {\partial A_k} {\partial x_k} + \sum_{k \mathop = 1}^n \frac {\partial U} {\partial x_k} A_k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map U {\nabla \cdot \mathbf A} + \sum_{k \mathop = 1}^n \frac {\partial U} {\partial x_k} A_k\) | Definition of Divergence Operator | |||||||||||
\(\ds \) | \(=\) | \(\ds \map U {\nabla \cdot \mathbf A} + \paren {\sum_{k \mathop = 1}^n \frac {\partial U} {\partial x_k} \mathbf e_k} \cdot \paren {\sum_{k \mathop = 1}^n A_k \mathbf e_k}\) | Definition of Dot Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \map U {\nabla \cdot \mathbf A} + \paren {\nabla U} \cdot \mathbf A\) | Definition of Gradient Operator, Definition of Vector |
$\blacksquare$
Also presented as
This result can also be presented as:
- $\nabla \cdot \paren {U \, \mathbf A} = \map U {\nabla \cdot \mathbf A} + \paren {\nabla U} \cdot \mathbf A$
presupposing the implementations of $\operatorname {div}$ and $\grad$ as operations using the del operator.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $7$. Divergence and Curl of $S \mathbf A$: $(4.13)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Miscellaneous Formulas involving $\nabla$: $22.38$