surface integral

The value of the surface integral ∬_S z dx dy where S is the external surface of the sphere x² + y² + z² = R² is

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...

Given a vector $\vec{u} = \frac{1}{3}(-y^3 \hat{i} + x^3 \hat{j} + z^3 \hat{k})$ and $\hat{n}$ as the unit normal vector to the surface of the hemisphere $(x^2 + y^2 + z^2 = 1; z \...

The surface integral $\iint_S \mathbf{F} \cdot \mathbf{n}\,dS$ over the surface $S$ of the sphere $x^2 + y^2 + z^2 = 9$, where $\mathbf{F}=(x+y)\mathbf{i}+(x+z)\mathbf{j}+(y+z)\mat...

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...