$Applications of Stokes' Theorem$

Question

$Applications of Stokes' Theorem$

Let $S$ be a surface and $C$ be a closed curve which is the boundary of $S$, and $f,g$ are $C^2$ functions. Show that

(i) $\int_C f \nabla g \cdot ds=\iint_S (\nabla f \times \nabla g)\cdot ds$
(ii) $\int_C (f \nabla g+g \nabla f)\cdot ds=0$

Multivariable Calculus Stokes' Theorem Integrals

Rosemary Green

13

Report

Answer

The answer is accepted.

Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

Answer 1

Answers can only be viewed under the following conditions:

The questioner was satisfied with and accepted the answer, or
The answer was evaluated as being 100% correct by the judge.

View the answer

(i) By Stokes' Theorem,
\[\int_{C} f\nabla g \cdot ds=\iint_{S} \nabla \times (f \nabla g)\cdot dS=\iint_S \left( \nabla f \times \nabla g+f (\nabla \times \nabla g) \right)\cdot dS\]
\[=\iint_S \left( \nabla f \times \nabla g\right)\cdot dS,\]
because $\nabla \times \nabla g=0.$

(ii) Again using Stokes' Theorem we have

\[\iint_C(f\nabla g+g\nabla f)\cdot ds=\int_C\nabla (fg)\cdot ds=\iint_S\nabla \times (\nabla (fg)) \cdot dS=0,\]
since $\nabla \times \nabla \phi=0$, for any function $\phi$.

Erdos

4.7K

$Applications of Stokes' Theorem$

Answer

Related Questions

Search