Confused on vector analysis notation: ($\cdot \mid \cdot)$

Question

Which of the following statements for scalar fields $Φ$ and $Ψ$ is correct? Justify the correctness of the true statements.

1. $\operatorname{div}\color{red}{\big(}\phi \operatorname{grad} (\psi)\color{red}{\mid}\psi \operatorname{grad} (\phi)\color{red}{\big)}=\phi \operatorname{div} (\operatorname{grad}(\psi))-\psi \operatorname{div} ( \operatorname{grad} (\phi))$

2. $\operatorname{\Delta} (\phi \operatorname{grad}(\psi)-\psi \operatorname{grad}(\phi))=\phi \operatorname{div}(\psi)-\psi \operatorname{div}(\psi)$

3. $\operatorname{div}(\phi \operatorname{grad}(\psi)-\psi \operatorname{grad}(\phi))=\phi \operatorname{\Delta} \psi - \psi \operatorname{\Delta} \phi $

4. $\color{red}{\big(}\operatorname{\nabla} \color{red}{\mid} \phi \operatorname{grad}(\psi)-\psi \operatorname{grad} (\phi)\color{red}{\big)}=\phi(\operatorname{\nabla} \times \psi)-\psi(\operatorname{\nabla} \times \phi)$

What I did:

2. false: $\operatorname{\Delta} \cdot \operatorname{grad} f\neq \operatorname{div}(f)$

3. false: $\operatorname{div}(\operatorname{grad}f)=\operatorname{\nabla} f\neq \operatorname{\Delta} f$

Questions: Can someone tell me is this correct and help me with the 1. and 4.?

I dont understand what exactly $|$ means here. I find that $|$ could mean "such that", "restricted of".. But I don't know how to use it here.

Answer 1

Assuming your professor/ book is using the notation $(\cdot \mid \cdot)$ to mean the inner product:

1. On the LHS is the divergence of the inner product of two vector fields. But the inner product of two vector fields is a scalar field and one can't take the divergence of a scalar field. Thus without even looking at the other side this can't be a correct identity because the LHS is undefined.
2. The RHS now has the divergence of a scalar field and is thus undefined.
3. This one is correct. $$\require{cancel}\operatorname{div}(\phi \operatorname{grad}(\psi)-\psi \operatorname{grad}(\phi))=\phi\operatorname{\Delta}\psi +\color{red}{\cancel{\color{black}{(\operatorname{grad}(\phi)\mid \operatorname{grad}(\psi))}}} - \psi\operatorname{\Delta}\phi -\color{red}{\cancel{\color{black}{(\operatorname{grad}(\psi)\mid \operatorname{grad}(\phi))}}}= \phi \operatorname{\Delta} \psi - \psi \operatorname{\Delta} \phi$$
4. The LHS is just an uncommon way of writing the divergence of a vector field and is thus a scalar field. On the RHS however you have a vector field. Thus LHS and RHS are different types of objects and cannot be equal.