Question

For a vector field \(\vec F\), the divergence theorem states that

• A. \(\displaystyle \iint_S \vec F\cdot d\vec s=\iiint_V \nabla\cdot\vec F\,dV\)
• B. \(\displaystyle \iint_S \vec F\cdot d\vec s=\iiint_V \nabla\times\vec F\,dV\)
• C. \(\displaystyle \iint_S \vec F\times d\vec s=\iiint_V \nabla\cdot\vec F\,dV\)
• D. \(\displaystyle \iint_S \vec F\times d\vec s=\iiint_V \nabla\times\vec F\,dV\)

Hint:

Gauss divergence theorem changes surface flux into a volume integral of divergence.

Answer 1

The Correct Option is A

Concept:
The divergence theorem is also called Gauss theorem. It relates the flux of a vector field through a closed surface to the volume integral of its divergence.

Step 1: Write the theorem. \[ \iint_S \vec F\cdot d\vec s = \iiint_V \nabla\cdot\vec F\,dV \] where \(S\) is a closed surface enclosing volume \(V\).

Step 2: Final answer. \[ \boxed{\iint_S \vec F\cdot d\vec s=\iiint_V \nabla\cdot\vec F\,dV} \]