( i) div(curl vec G )=0. (ii) curl( nabla f)= vec 0 . (iii) curl(f vec G )=f(curl vec G )+( nabla f)*( vec G ). (iv) div( nabla f)= nabla^ 2 f. (v) div( vec F + vec G )=div vec F +div vec G . (vi) div (f vec G )=f(div vec G )+( nabla f)*( vec G ). (vii) nabla(fg)=f nabla g+g nabla f.

Rewrite the formulas (i)-(vi) from problem 3 using the notations nabla* vec H instead of div vec H and nabla* vec H instead of curl vec H (For example , (iv ) can be written as nabla*( nabla f)= nabla^ 2 f.)

Let's rewrite the formulas using the new notation \( \nabla^* \) in place of both \( \text{div} \) and \( \text{curl} \). Based on your instructions, the \( \nabla^* \) will now represent these two different operations depending on the context. Here's the rewritten set of equations:

1. (i) \( \nabla^* (\nabla^* \vec{G}) = 0 \)

2. (ii) \( \nabla^* (\nabla f) = \vec{0} \)

3. (iii) \( \nabla^* (f \vec{G}) = f (\nabla^* \vec{G}) + (\nabla f) \cdot \vec{G} \)

4. (iv) \( \nabla^* (\nabla f) = \nabla^2 f \)

5. (v) \( \nabla^* (\vec{F} + \vec{G}) = \nabla^* \vec{F} + \nabla^* \vec{G} \)

6. (vi) \( \nabla^* (f \vec{G}) = f (\nabla^* \vec{G}) + (\nabla f) \cdot \vec{G} \)

Notice that equations (iii) and (vi) end up looking identical because they both involve the product of a scalar function \( f \) and a vector field \( \vec{G} \). In both cases, \( \nabla f \) is the gradient of \( f \), and \( \cdot \) indicates a dot product between \( \nabla f \) and the vector \( \vec{G} \).

Would you like further details on any of these steps, or have any questions?

Here are 5 related questions to explore further:
1. How does the divergence of a vector field relate to the curl of the same field?
2. Can you provide a physical interpretation of \( \nabla^* \) in the context of fluid dynamics?
3. What is the difference between the Laplacian \( \nabla^2 f \) and the gradient \( \nabla f \)?
4. How does the product rule apply in vector calculus, as seen in equation (vii)?
5. What are some common applications of these vector calculus identities in physics?

**Tip:** When dealing with vector calculus, always keep in mind that \( \nabla \) behaves differently when applied to scalars vs. vectors.

Rewrite the formulas (i)-(vi) from problem 3 using the notations nabla* vec H instead of div vec H and nabla* vec H instead of curl vec H.

This guide explains how to rewrite common vector calculus formulas using the notation nabla* for divergence and curl operations. The rewritten formulas cover concepts like divergence, gradient, and Laplacian. Suitable for students studying vector calculus at the undergraduate level.

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