Question

$\mathbf{\iint }\mathbf{(}\mathbf{2}\mathbf{xi}\mathbf{-}\mathbf{2}\mathbf{yj}\mathbf{+}\mathbf{5}\mathbf{k}\mathbf{)}\mathbf{\times }\mathbf{nd\sigma }$over the surface of a sphere of radius $\mathbf{2}$and center at the origin.

Short answer

The solution derived is $\mathrm{I}=0$.

Step-by-step solution

Given Information.

The given expression is$\iint (2\mathrm{xi}-2\mathrm{yj}+5\mathrm{k})\times \mathrm{nd\sigma }$.

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines its magnitude.

Apply divergence theorem.

Apply the divergence theorem and the fact that $\mathrm{\nabla }\times \mathrm{V}=2-2=0$then the integral becomes as shown below.

$\mathrm{I}=0\times {\iiint }_{\mathrm{V}} \mathrm{d\tau }$

$=0$

Hence, the solution derived is $\mathrm{I}=0$.