Question

Math

Posted 5 months ago

```
Assume that $S$ is an inwardly oriented, piecewise-smooth surface with a piecewise-smooth, simple, closed boundary curve $C$ oriented negatively with respect to the orientation of $S$.
Let $F$ be a continuously differentiable vector field in $\mathbb{R}^{3}$ such that $F$ equals 0 whenever $z=0$.
Does Stokes' theorem necessarily apply to the surface $S$, boundary curve $C$, and vector field $F ?$
Choose 1 answer:
(A) Yes
(B) No
```

Sign in to unlock the answer

Answer from Sia

Posted 5 months ago

Solution by Steps

step 2

The condition that $F$ equals 0 whenever $z = 0$ does not violate the requirements for Stokes' theorem. The theorem does not require the vector field to be non-zero everywhere

step 3

Since $S$ is inwardly oriented with a negatively oriented boundary $C$ with respect to $S$, and $F$ is continuously differentiable, Stokes' theorem applies to the given surface $S$, boundary curve $C$, and vector field $F$

1 Answer

A

Key Concept

Stokes' theorem applicability

Explanation

Stokes' theorem applies to any piecewise-smooth surface with a piecewise-smooth boundary curve, provided the vector field is continuously differentiable over the surface and its boundary, regardless of the vector field's values on a particular plane.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question