Use Stokes' Theorem to Calculate the Circulation of the Field $\mathbf { F } = - 9 \mathrm { y } ^ { 3 } \mathbf { i } + 9 \mathrm { x } ^ { 3 } \mathbf { j } + 8 \mathrm { z } ^ { 3 } \mathbf { k }$

question 275

Multiple Choice

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction.
- $\mathbf { F } = - 9 \mathrm { y } ^ { 3 } \mathbf { i } + 9 \mathrm { x } ^ { 3 } \mathbf { j } + 8 \mathrm { z } ^ { 3 } \mathbf { k }$ ; C: the portion of the paraboloid $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = \mathrm { z }$ cut by the cylinder $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$

• A) \(- 432 \pi\)
• B) \(- 216 \pi\)
• C) \(432 \pi\)
• D) \(216 \pi\)

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Definitions:

Fraternal Twins

Twins formed from the fertilization of two separate eggs by two different sperm cells, resulting in genetically distinct individuals.

Monozygotic

Refers to twins that are genetically identical, resulting from a single fertilized egg splitting into two.

Identical Twins

Twins who are genetically identical, having developed from a single fertilized egg that split into two, sharing the same DNA.

Epigenetic

Referring to heritable changes in gene expression that do not involve changes to the underlying DNA sequence.