Using Either Stokes' Theorem or the Divergence Theorem (Whichever Is $\mathrm { Q } ( - 4 x \vec { i } + 3 y \vec { j } + 3 z \vec { k } ) \times \vec { d r }$

question 26

Short Answer

Using either Stokes' theorem or the Divergence theorem (whichever is appropriate), evaluate the following: $\mathrm { Q } ( - 4 x \vec { i } + 3 y \vec { j } + 3 z \vec { k } ) \times \vec { d r }$ where C is a closed loop parameterized by $\vec { r } ( t ) = ( \cos t + \sin t ) \vec { i } + \cos ^ { 3 } t \vec { j } + \sin ^ { 4 } t \vec { k }$

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Definitions:

Sunk Cost

A cost that has already been incurred and cannot be recovered, and therefore should not affect current and future business decisions.

Nonrefundable Ticket

A type of ticket for which the cost is not returned to the purchaser if they decide to cancel their booking or are unable to use the ticket.

Soccer Match

A competitive game of soccer played between two teams, adhering to specific rules and durations as specified by the sport's governing body.

Average Fixed Costs

The fixed costs (expenses that do not change with the level of output) divided by the quantity of output produced. It decreases as output increases.