On an Exam, Students Are Asked to Find the Line ${ \vec { F } } = 2 x y \vec { i } + y z \vec { j } + y ^ { 2 } \vec { k }$

question 7

Multiple Choice

On an exam, students are asked to find the line integral of ${ \vec { F } } = 2 x y \vec { i } + y z \vec { j } + y ^ { 2 } \vec { k }$ over the curve C which is the boundary of the upper hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 ^ { 2 } , z \geq 0$ oriented in a counter-clockwise direction when viewed from above.One student wrote: " $\operatorname { curl } { \vec { F } } = - 2 x \vec { j } + 2 \vec { k }$ By Stokes' Theorem $\int _ { C } \vec { F } \cdot \vec{ d r } = \int _ { S } ( - 2 x \vec { j } + 2 \vec { k } ) \cdot \vec { d A }$ where S is the hemisphere.Since $\operatorname { div } ( - 2 x \vec { j } + 2 \vec { k } ) = 0$ by the Divergence Theorem $\int _ { S } ( - 2 x \vec { j } + 2 \vec { k } ) \cdot \vec { d A } = \int _ { W } 0 d V = 0$ where W is the solid hemisphere.Hence we have $\int _ { C } \vec { F } \cdot \vec{ d r } = 0$ "
This answer is wrong.Which part of the student's argument is wrong? Select all that apply.

• A)The student has been careless with the orientations of the curve and surface.
• B)The student has to be careful with the orientations of the curve and surface.However, Stokes' Theorem has been applied correctly.
• C)The student has used Divergence Theorem incorrectly.The upper hemisphere S does not include the bottom, hence it does not enclose any region in space, and we cannot apply the Divergence Theorem to this surface.
• D)The student has to be careful with the orientations of the curve and surface.However, the Divergence Theorem has been applied correctly.
• E)The student has the correct orientations of the curve and surface but the student has used Divergence Theorem for the hemisphere incorrectly.The upper hemisphere S does not include the bottom, hence it does not enclose any region in space, and we cannot apply the Divergence Theorem to this surface.

---

Definitions:

Procedural Error

A mistake made during a legal process that may impact the fairness or validity of the proceedings or outcome.

Intermediate Appellate Courts

Courts that review and decide on appeals from lower courts, serving as an intermediary between trial courts and the highest court in a jurisdiction.

Appellate Briefs

Written documents submitted by parties in an appellate court, outlining arguments and legal theories for overturning or upholding a lower court’s decision.

Oral Arguments

The process in a legal proceeding where attorneys present their case verbally before a judge or panel, emphasizing key points.