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:

Batch Setup

Batch setup involves the processes and activities necessary to prepare equipment and facilities for the production of a batch of goods.

Human Resource Management

The strategic approach to the effective management of people in an organization to help the business gain a competitive advantage.

Facility-level Activity

Activities and costs incurred that support the entire company and cannot be traced to individual products or services.

Product-level Activity

Actions or operations that are conducted to support or produce specific products, often analyzed in activity-based costing.