Question

Prove each identity, assuming that \({\rm{S}}\) and \({\rm{E}}\) satisfy the conditions of the Divergence Theorem and the scalar function and components of the vector fields have continuous second-order partial derivatives. \(\iint_{\text{S}} {{\text{curl F}} \times {\text{dS = 0}}}\).

Short answer

It is proved, \(\iint_{\text{S}} {{\text{curl F}} \times {\text{dS = 0}}}\).

Step-by-step solution

Define vector

A vector is a number that specifies not only the size of an item, but also its movement or location with respect to another point or object.

Explanation

The divergence theorem states that,

\(\iint_{\text{S}} {{\text{F}} \times {\text{dS = }}\iiint_{\text{E}} {{\text{div(F)dV}}}}\)

Where, \({\rm{S}}\)= closed surface

\({\rm{E}}\)= inside that surface's region

One may state using the divergence theory that,

\(\begin{gathered} \iint_{\text{S}} {{\text{curl F}} \times {\text{ds}}}{\text{ = }}\iiint_{\text{E}} {{\text{div (curl F)}}}{\text{dV}} \\ {\text{ = }}\iiint_{\text{E}} {\text{0}}{\text{dV}} \\ {\text{ = 0}} \\ \end{gathered} \)

It takes use of the fact that,

\({\rm{div (curl F) = 0}}\)

Therefore,\(\iint_{\text{S}} {{\text{curl F}} \times {\text{dS = 0}}}\) .