Prove the product rules a. ∇ · (A \times B) = B · (∇ \times A) − A · (∇ \times B) b. ∇ \times (fA) = f(∇ \times A) − A \times (∇f)
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Part (a): Prove the product rule for the divergence of a cross product ** Show more…
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Let $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ be differentiable vector fields, and let $a$ and $b$ be arbitrary real constants. Verify the following identities. a. $\nabla \cdot\left(a \mathbf{F}_{1}+b \mathbf{F}_{2}\right)=a \nabla \cdot \mathbf{F}_{1}+b \nabla \cdot \mathbf{F}_{2}$ b. $\nabla \times\left(a \mathbf{F}_{1}+b \mathbf{F}_{2}\right)=a \nabla \times \mathbf{F}_{1}+b \nabla \times \mathbf{F}_{2}$ c. $\nabla \cdot\left(\mathbf{F}_{1} \times \mathbf{F}_{2}\right)=\mathbf{F}_{2} \cdot \nabla \times \mathbf{F}_{1}-\mathbf{F}_{1} \cdot \nabla \times \mathbf{F}_{2}$
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If $\mathbf{F}=M \mathbf{i}+N \mathbf{j}+P \mathbf{k}$ is a differentiable vector field, we define the notation $\mathbf{F} \cdot \nabla$ to mean $$M \frac{\partial}{\partial x}+N \frac{\partial}{\partial y}+P \frac{\partial}{\partial z}$$ For differentiable vector fields $\mathbf{F}_{1}$ and $\mathbf{F}_{2},$ verify the following identities. a. $\nabla \times\left(\mathbf{F}_{1} \times \mathbf{F}_{2}\right)=\left(\mathbf{F}_{2} \cdot \nabla\right) \mathbf{F}_{1}-\left(\mathbf{F}_{1} \cdot \nabla\right) \mathbf{F}_{2}+\left(\nabla \cdot \mathbf{F}_{2}\right) \mathbf{F}_{1}-$ $\left(\nabla \cdot \mathbf{F}_{1}\right) \mathbf{F}_{2}$ b. $\nabla\left(\mathbf{F}_{1} \cdot \mathbf{F}_{2}\right)=\left(\mathbf{F}_{1} \cdot \nabla\right) \mathbf{F}_{2}+\left(\mathbf{F}_{2} \cdot \nabla\right) \mathbf{F}_{1}+\mathbf{F}_{1} \times\left(\nabla \times \mathbf{F}_{2}\right)+$ $\mathbf{F}_{2} \times\left(\nabla \times \mathbf{F}_{1}\right)$
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