## Proof of Identity for Differences of Cross Products of Vectors With Curls

Theorem>
If
$\mathbf{v}$
and
$\mathbf{w}$
are vectors then
$\mathbf{v} \times (\mathbf{\nabla} \times \mathbf{w}) - (\mathbf{w} \times \mathbf{\nabla}) \times \mathbf{v} = \mathbf{v} (\mathbf{\nabla} \cdot \mathbf{w}) - (\mathbf{v} \cdot \mathbf{\nabla}) \mathbf{w}$

Proof
Let
$\mathbf{v}=v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k}$
and
$\mathbf{w}=w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}$

\begin{aligned} \mathbf{v} \times (\mathbf{\nabla} \times \mathbf{w}) - (\mathbf{w} \times \mathbf{\nabla}) \times \mathbf{v} &= (v_1 \mathbf{i} + v_2 \mathbf{j} + v_3) \times ((\frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k}) \times (w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}) \\ &- ((v_1 \mathbf{i} + v_2 \mathbf{j} + v_3) \times (\frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k})) \times (w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}) \\ &= (v_1 \mathbf{i} + v_2 \mathbf{j} + v_3) \times ((\frac{\partial w_3}{\partial y} - \frac{\partial w_2}{\partial z}) \mathbf{i} + (\frac{\partial w_1}{\partial z} - \frac{\partial w_3}{\partial x}) \mathbf{j} + (\frac{\partial w_2}{\partial x} - \frac{\partial w_1}{\partial y}) \mathbf{k} ) \\ &- ((v_2 \frac{\partial}{\partial z} - v_3 \frac{\partial}{\partial y}) \mathbf{i} + (v_3 \frac{\partial}{\partial x} - v_1 \frac{\partial }{\partial z}) \mathbf{j} + (v_1 \frac{\partial }{\partial y} - v_2 \frac{\partial }{\partial x}) \mathbf{k} ) \times(w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}) \\ &= (v_2 (\frac{\partial w_2}{\partial x} - \frac{\partial w_1}{\partial y}) - v_3 (\frac{\partial w_1}{\partial z} - \frac{\partial w_3}{\partial x})) \mathbf{i} \\ &+ (v_3 (\frac{\partial w_3}{\partial y} - \frac{\partial w_2}{\partial z}) -v_1 (\frac{\partial w_2}{\partial x} - \frac{\partial w_1}{\partial y})) \mathbf{j} \\ &+ (v_1 (\frac{\partial w_1}{\partial z} - \frac{\partial w_3}{\partial x}) -v_2 (\frac{\partial w_3}{\partial y} - \frac{\partial w_2}{\partial z}) ) \mathbf{k} \\ &- ((v_3 \frac{\partial w_3}{\partial x} - v_1 \frac{\partial w_3}{\partial z}) - (v_1 \frac{\partial w_2}{\partial y} - v_2 \frac{\partial w_2}{\partial x})) \mathbf{i} \\ &- ((v_1 \frac{\partial w_1}{\partial y} - v_2 \frac{\partial w_1}{\partial x}) - (v_2 \frac{\partial w_3}{\partial z} - v_3 \frac{\partial w_3}{\partial y})) \mathbf{j} \\ &- ((v_2 \frac{\partial w_2}{\partial z} - v_3 \frac{\partial w_2}{\partial y}) - (v_3 \frac{\partial w_1}{\partial x} - v_1 \frac{\partial w_1}{\partial z})) \mathbf{k} \\ &= v_1 (\frac{\partial w_1}{\partial x} \mathbf{i} + \frac{\partial w_2}{\partial y} \mathbf{j} + \frac{\partial w_3}{\partial z} \mathbf{k} ) \\ &+ v_2 (\frac{\partial w_1}{\partial x} \mathbf{i} + \frac{\partial w_2}{\partial y} \mathbf{j} + \frac{\partial w_3}{\partial z} \mathbf{k} ) \\ &+ v_3 (\frac{\partial w_1}{\partial x} \mathbf{i} + \frac{\partial w_2}{\partial y} \mathbf{j} + \frac{\partial w_3}{\partial z} \mathbf{k} ) \\ &- v_1 \frac{\partial}{\partial x} (w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}) \\ &- v_2 \frac{\partial}{\partial x} (w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}) \\ &- v_3 \frac{\partial}{\partial x} (w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}) \\ &= \mathbf{v} (\mathbf{\nabla} \cdot \mathbf{w}) - (\mathbf{v} \cdot \mathbf{\nabla}) \mathbf{w} \end{aligned}