Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point (1, 2, 3). The surface integral $\int_A \vec{F}.d\vec{A}$ of a vector field $\vec{F}=3x\hat{i}+5y\hat{j}+6z\hat{k}$ over the entire surface A of the cube is ______.

Given a function $\rm ϕ = \frac{1}{2} (x^2 + y^2 + z^2) $ in three-dimensional Cartesian space, the value of the surface integral

∯_S n̂ . ∇ϕ dS

where S is the surface of a sphere of unit radius and n̂ is the outward unit normal vector on S, is

For three vectors $$\vec A = 2\hat j - 3\hat k,\vec B = - 2\hat i + \hat k\ and\;\vec C = 3\hat i - \hat j,$$ where î, ĵ and k̂ are unit vectors along the axes of a right-handed rectangular/Cartesian coordinate system, the value of $$\left( {\vec {A.} \left( {\vec B \times \vec C} \right) + 6} \right)$$ is _______.

Let $$\phi $$ be an arbitrary smooth real valued scalar function and $$\overrightarrow V $$ be an arbitrary smooth vector valued function in a three dimensional space. Which one of the following is an identity?