The velocity field on an incompressible flow is given by
$$V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)j\,$$ $$ + \left( {{c_1}x + {c_2}y + {c_3}z} \right)k,\,\,$$
where $${{a_1} = 2}$$ and $${{c_3} = - 4.}$$ The value of $${{b_2}}$$ is ________.

The integral $$\,\,\oint\limits_C {\left( {ydx - xdy} \right)\,\,} $$ is evaluated along the circle $${x^2} + {y^2} = {1 \over 4}\,$$ traversed in counter clockwise direction. The integral is equal to

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field $$F = xi + yj + zk$$ defined with respect to a Cartesian coordinate system having $$i, j$$ and $$k$$ as unit base vectors. $$$\int {\int\limits_S {{1 \over 4}\left( {F.n} \right)dA} } $$$

Where $$S$$ is the sphere, $$\,\,{x^2} + {y^2} + {z^2} = 1\,\,$$ and $$n$$ is the outward unit normal vector to the sphere. The value of the surface integral is

The divergence of the vector field $$\,3xz\widehat i + 2xy\widehat j - y{z^2}\widehat k$$ at a point $$(1,1,1)$$ is equal to