Let \[{T_{ij}} = \sum\limits_K {{\varepsilon _{ijk}}{a_k}} \]    and \[{\beta _k} = \sum\limits_{i,\,j} {{\varepsilon _{ijk}}{T_{ij}}} ,\]    where \[{\varepsilon _{ijk}}\]  is the Levi-Civita density, defined to be zero, if two 'of the indices. coincide and +1 and -1 depending on whether ijk is even or odd permutation of 1, 2, 3. Then β₃ is equal to

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