Find the Flux of the Curl of Field F Through $\mathbf { F } = ( x - y ) \mathbf { i } + ( x - z ) \mathbf { j } + ( y - z ) \mathbf { k } ; S$

question 18

Multiple Choice

Find the flux of the curl of field F through the shell S.
- $\mathbf { F } = ( x - y ) \mathbf { i } + ( x - z ) \mathbf { j } + ( y - z ) \mathbf { k } ; S$ is the portion of the cone $z = 3 \sqrt { x ^ { 2 } + y ^ { 2 } }$ below the plane $z = 2$

• A) \(- \frac { 8 } { 9 } \pi\)
• B) \(\frac { 8 } { 9 } \pi\)
• C) \(\frac { 16 } { 9 }\)
• D) \(- \frac { 16 } { 9 } \pi\)

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Definitions:

Marginal Revenue

The additional revenue that a firm receives from selling one more unit of a good or service.

Average Total Cost

The total cost of production divided by the total output or quantity produced.

Monopolistically Competitive Firm

A firm that operates in a market with many competitors, each offering a slightly different product.

Profit-Maximizing Quantity

The level of production at which a firm achieves the highest possible profit, where marginal revenue equals marginal cost.