To find $\mathbf{a} \times (\nabla \times \mathbf{b}) + \mathbf{b} \times (\nabla \times \mathbf{a})$, we can apply a vector calculus identity known as the vector triple product identity. Here's how it works:

Step-by-Step Solution:

1. We start with the expression: $\mathbf{a} \times (\nabla \times \mathbf{b}) + \mathbf{b} \times (\nabla \times \mathbf{a})$

2. This can be simplified using a known vector identity: $\mathbf{a} \times (\nabla \times \mathbf{b}) + \mathbf{b} \times (\nabla \times \mathbf{a}) = \nabla (\mathbf{a} \cdot \mathbf{b}) - (\mathbf{a} \cdot \nabla)\mathbf{b} - (\mathbf{b} \cdot \nabla)\mathbf{a}$

Explanation:

• $\nabla (\mathbf{a} \cdot \mathbf{b})$ represents the gradient of the dot product of $\mathbf{a}$ and $\mathbf{b}$, which is a scalar field.
• The terms $(\mathbf{a} \cdot \nabla)\mathbf{b}$ and $(\mathbf{b} \cdot \nabla)\mathbf{a}$ are directional derivatives. They involve taking the derivative of one vector field in the direction of the other.

Thus, the full simplified expression is: $\boxed{\nabla (\mathbf{a} \cdot \mathbf{b}) - (\mathbf{a} \cdot \nabla)\mathbf{b} - (\mathbf{b} \cdot \nabla)\mathbf{a}}$

This identity is very useful in various physics and engineering contexts, especially in electromagnetism and fluid dynamics.

---

Do you want more details or have any questions?

Here are 5 related questions that you might find useful:

1. How do we compute the curl of a vector field?
2. What are some other vector identities in vector calculus?
3. How is the vector triple product used in physics?
4. Can you explain directional derivatives in more detail?
5. How do these vector operations relate to physical quantities in electromagnetism?

Tip: When dealing with vector identities, always check if standard identities like the triple product or divergence/curl identities can simplify the problem.