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

Question

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

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Stokes' theorem can be applied to any piecewise-smooth surface $ S $ with a piecewise-smooth, simple, closed boundary curve $ C $, provided the vector field $ F $ is continuously differentiable over the surface and its boundary. ‖ ‖ 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.⚹◊What is the mathematical representation of Stokes' theorem in terms of a surface integral and a line integral?⍭ Generate me a similar question◊

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