Divergence of Vector Cross Product
Definition
Let $R$ be a region of space embedded in Cartesian $3$ space $\R^3$.
Let $\mathbf A$ and $\mathbf B$ be vector fields over $R$.
Then:
- $\map {\operatorname {div} } {\mathbf A \times \mathbf B} = \mathbf B \cdot \curl \mathbf A - \mathbf A \cdot \curl \mathbf B$
where:
- $\operatorname {div}$ denotes the divergence operator
- $\curl$ denotes the curl operator
- $\times$ denotes vector cross product
- $\cdot$ denotes dot product.
Proof
From Divergence Operator on Vector Space is Dot Product of Del Operator and Curl Operator on Vector Space is Cross Product of Del Operator:
\(\ds \operatorname {div} \mathbf V\) | \(=\) | \(\ds \nabla \cdot \mathbf V\) | ||||||||||||
\(\ds \curl \mathbf V\) | \(=\) | \(\ds \nabla \times \mathbf V\) |
where $\nabla$ denotes the del operator.
Hence we are to demonstrate that:
- $\nabla \cdot \paren {\mathbf A \times \mathbf B} = \mathbf B \cdot \paren {\nabla \times \mathbf A} - \mathbf A \cdot \paren {\nabla \times \mathbf B}$
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf A$ and $\mathbf B: \R^3 \to \R^3$ be expressed as vector-valued functions on $\R^3$:
- $\mathbf A := \tuple {\map {A_x} {\mathbf r}, \map {A_y} {\mathbf r}, \map {A_z} {\mathbf r} }$
- $\mathbf B := \tuple {\map {B_x} {\mathbf r}, \map {B_y} {\mathbf r}, \map {B_z} {\mathbf r} }$
where $\mathbf r = \tuple {x, y, z}$ is the position vector of an arbitrary point in $R$.
Then:
\(\ds \nabla \cdot \paren {\mathbf A \times \mathbf B}\) | \(=\) | \(\ds \nabla \cdot \paren {\paren {A_y B_z - A_z B_y} \mathbf i + \paren {A_z B_x - A_x B_z} \mathbf j + \paren {A_x B_y - A_y B_x} \mathbf k}\) | Definition 1 of Vector Cross Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\dfrac \partial {\partial x} } {A_y B_z - A_z B_y} + \map {\dfrac \partial {\partial y} } {A_z B_x - A_x B_z} + \map {\dfrac \partial {\partial z} } {A_x B_y - A_y B_x}\) | Definition of Divergence Operator | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac {\partial A_y B_z} {\partial x} - \dfrac {\partial A_z B_y} {\partial x} } + \paren {\dfrac {\partial A_z B_x} {\partial y} - \dfrac {\partial A_x B_z} {\partial y} } + \paren {\dfrac {\partial A_x B_y} {\partial z} - \dfrac {\partial A_y B_x} {\partial z} }\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {A_y \dfrac {\partial B_z} {\partial x} + \dfrac {\partial A_y} {\partial x} B_z - A_z \dfrac {\partial B_y} {\partial x} - \dfrac {\partial A_z} {\partial x} B_y} + \paren {A_z \dfrac {\partial B_x} {\partial y} + \dfrac {\partial A_z} {\partial y} B_x - A_x \dfrac {\partial B_z} {\partial y} - \dfrac {\partial A_x} {\partial y} B_z} + \paren {A_x \dfrac {\partial B_y} {\partial z} + \dfrac {\partial A_x} {\partial z} B_y - A_y \dfrac {\partial B_x} {\partial z} - \dfrac {\partial A_y} {\partial z} B_x}\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {B_x} {\dfrac {\partial A_z} {\partial y} - \dfrac {\partial A_y} {\partial z} } + \map {B_y} {\dfrac {\partial A_x} {\partial z} - \dfrac {\partial A_z} {\partial x} } + \map {B_z} {\dfrac {\partial A_y} {\partial x} - \dfrac {\partial A_x} {\partial y} }\) | rearranging | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \map {A_x} {\dfrac {\partial B_y} {\partial z} - \dfrac {\partial B_z} {\partial y} } + \map {A_y} {\dfrac {\partial B_z} {\partial x} - \dfrac {\partial B_x} {\partial z} } + \map {A_z} {\dfrac {\partial B_x} {\partial y} - \dfrac {\partial B_y} {\partial x} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {B_x} {\dfrac {\partial A_z} {\partial y} - \dfrac {\partial A_y} {\partial z} } + \map {B_y} {\dfrac {\partial A_x} {\partial z} - \dfrac {\partial A_z} {\partial x} } + \map {B_z} {\dfrac {\partial A_y} {\partial x} - \dfrac {\partial A_x} {\partial y} }\) | further rearranging | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \map {A_x} {\dfrac {\partial B_z} {\partial y} - \dfrac {\partial B_y} {\partial z} } + \map {A_y} {\dfrac {\partial B_x} {\partial z} - \dfrac {\partial B_z} {\partial x} } + \map {A_z} {\dfrac {\partial B_y} {\partial x} - \dfrac {\partial B_x} {\partial y} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {B_x \mathbf i + B_y \mathbf j + B_z \mathbf k} \cdot \paren {\paren {\dfrac {\partial A_z} {\partial y} - \dfrac {\partial A_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial A_x} {\partial z} - \dfrac {\partial A_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial A_y} {\partial x} - \dfrac {\partial A_x} {\partial y} } \mathbf k}\) | Definition of Dot Product | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \paren {A_x \mathbf i + A_y \mathbf j + A_z \mathbf k} \cdot \paren {\paren {\dfrac {\partial B_z} {\partial y} - \dfrac {\partial B_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial B_x} {\partial z} - \dfrac {\partial B_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial B_y} {\partial x} - \dfrac {\partial B_x} {\partial y} } \mathbf k}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf B \cdot \paren {\nabla \times \mathbf A} - \mathbf A \cdot \paren {\nabla \times \mathbf B}\) | Definition of Curl Operator |
$\blacksquare$
Also presented as
This result can also be presented as:
- $\nabla \cdot \paren {\mathbf A \times \mathbf B} = \mathbf B \cdot \paren {\nabla \times \mathbf A} - \mathbf A \cdot \paren {\nabla \times \mathbf B}$
presupposing the implementations of $\operatorname {div}$ and $\curl$ 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: $6$. Divergence of a Vector Product: $(4.12)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Miscellaneous Formulas involving $\nabla$: $22.40$