Can the following scalar and vector potentials describe an electromagnetic field? \[\phi \left( {\overrightarrow x ,\,t} \right) = 3xyz - 4t\]
\[\overrightarrow A \left( {\overrightarrow x ,\,t} \right) = \left( {2x - \omega t} \right)\hat i + \left( {y - 2z} \right)\hat j + \left( {z - 2{e^{i\omega t}}} \right)\hat k\]
where, ω is a constant.

The contour integral $$\oint {\frac{{dz}}{{{z^2} + {a^2}}}} $$   is to be evaluated on a circle of radius 2a centred at the origin. It will have contributions only from the points

Which of the following functions of the complex variable z is not analytic everywhere?

Consider a vector \[\overrightarrow {\bf{p}} = 2{\bf{\hat i}} + 3{\bf{\hat j}} + 2{\bf{\hat k}}\]     in the coordinate system \[\left( {{\bf{\hat i}},\,{\bf{\hat j}},\,{\bf{\hat k}}} \right).\]   The axes are rotated anti-clockwise about the Y-axis by an angle of 60°. The vector \[\overrightarrow p \] in the rotate coordinate system \[\left( {{\bf{\hat i}},\,{\bf{\hat j}},\,{\bf{\hat k}}} \right)\]   is

The value of the integral $$\oint\limits_C {\frac{{{e^z}\sin z}}{{{z^2}}}} dz,$$   where the contour C is the unit circle: |z - 2| = 1, is