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Qmechanic ♦
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Vector triple product with del$\nabla$ operator

I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):

$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\mathbf{B} \cdot \mathbf{\nabla} \right)\mathbf{B}- \frac{1}{2}\nabla B^2 $$

But, if I use the BAC-CAB rule for this triple product operation I get: $$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\mathbf{B} \cdot \mathbf{\nabla} \right)\mathbf{B}- \nabla B^2 $$

Which does not have the $1/2$ term on it. Do you know why is this?

Thanks!

Vector triple product with del operator

I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):

$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\mathbf{B} \cdot \mathbf{\nabla} \right)\mathbf{B}- \frac{1}{2}\nabla B^2 $$

But, if I use the BAC-CAB rule for this triple product operation I get: $$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\mathbf{B} \cdot \mathbf{\nabla} \right)\mathbf{B}- \nabla B^2 $$

Which does not have the $1/2$ term on it. Do you know why is this?

Thanks!

Vector triple product with $\nabla$ operator

I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):

$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\mathbf{B} \cdot \mathbf{\nabla} \right)\mathbf{B}- \frac{1}{2}\nabla B^2 $$

But, if I use the BAC-CAB rule for this triple product operation I get: $$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\mathbf{B} \cdot \mathbf{\nabla} \right)\mathbf{B}- \nabla B^2 $$

Which does not have the $1/2$ term on it. Do you know why is this?

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kid_a
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Vector triple product with del operator

I came across the following expression in several books (especially in plasma physics literature while deriving the magnetic pressure):

$$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\mathbf{B} \cdot \mathbf{\nabla} \right)\mathbf{B}- \frac{1}{2}\nabla B^2 $$

But, if I use the BAC-CAB rule for this triple product operation I get: $$(\mathbf{\nabla} \times \mathbf{B})\times \mathbf{B} = \left(\mathbf{B} \cdot \mathbf{\nabla} \right)\mathbf{B}- \nabla B^2 $$

Which does not have the $1/2$ term on it. Do you know why is this?

Thanks!