(Solved): 1 please give full answer for all questions! 1. (a) Using index notation, pro ...

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please give full answer for all questions!???????

1. (a) Using index notation, prove the identity $$???\times \mathbf{A}?0,$$ where $$\mathbf{A}$$ is a general vector field. (b) Using index notation, prove the identity $$??\left(\frac{\mathbf{F}}{g}\right)?\frac{g??\mathbf{F}?\mathbf{F}??g}{g^2}$$ where $$\mathbf{F}$$ and $$g$$ are general vector and scalar fields respectively. (c) By setting $$\mathbf{F}=?\times \mathbf{A}$$, use the identities in parts (a) and (b) to find an expression for $$\mathrm{div}(($$ curl $$\mathbf{A})/g)$$ that involves no second derivatives. (d) Given that general expressions for grad and curl are given, in cylindrical polar basis, by $$\begin{array}{rl}\mathrm{grad}g& ?\left(\frac{?g}{?R}\right){\mathbf{e}}_R+\frac{1}{R}\left(\frac{?g}{??}\right){\mathbf{e}}_{?}+\left(\frac{?g}{?z}\right){\mathbf{e}}_z,\\ & \mathrm{curl}\mathbf{B}?\frac{1}{R}\left|\begin{array}{ccc}{\mathbf{e}}_R& {\mathrm{Re}}_{?}& {\mathbf{e}}_z\\ \frac{?}{?R}& \frac{?}{??}& \frac{?}{?z}\\ B_R& R{B}_{?}& B_z\end{array}\right|\end{array}$$ find the curl of the vector field $$\mathbf{B}=R{\mathbf{e}}_{?}+z^2{\mathbf{e}}_z$$ and hence use the result of part (c) to evaluate the divergence of the field $$\frac{2{\mathbf{e}}_z}{R+z}$$