divergence theorem

A vector field B(x, y, z) = x î + y ĵ – 2z k is defined over a conical region having height h = 2, base radius r = 3 and axis along z, as shown in the figure. The base of the cone...

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...

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 fie...

The value of the integral $\oiint_S \vec{r} \cdot \vec{n} \, dS$ over the closed surface S bounding a volume V, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ is the position vec...

The surface integral $$\int {\int\limits_s {F.ndS} } $$ over the surface $$S$$ of the sphere $${x^2} + {y^2} + {z^2} = 9,$$ where $$\,F = \left( {x + y} \right){\rm I} + \left( {x...

The surface integral $$\,\,\int {\int\limits_s {{1 \over \pi }} } \left( {9xi - 3yj} \right).n\,dS\,\,$$ over the sphere given by $${x^2} + {y^2} + {z^2} = 9\,\,$$ is __________.

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 syste...