Find the Flux Is the Upward Unit Normal Vector to S

question 72

Multiple Choice

Find the flux $\overrightarrow { \mathrm { F } } \text { of through } S , \iint _ { S } \overrightarrow { \mathrm { F } } \cdot \overrightarrow { \mathrm { N } } d S \text {, where } \overrightarrow { \mathrm { N }}$ is the upward unit normal vector to S.
$\overrightarrow { \mathrm { F } } ( x , y , z ) = x \hat { \mathbf { i } } + y \hat { \mathbf { j } } + z \hat { \mathbf { k } }$
$S : z = 64 - x ^ { 2 } - y ^ { 2 } , z \geq 0$

• A) \(4,301 \pi\)
• B) \(6,144 \pi\)
• C) \(7,987 \pi\)
• D) \(3072 \pi\)
• E) \(1843.2 \pi\)

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Definitions:

Neutralize

To render something harmless by applying an opposite force or effect, such as neutralizing an acid with a base.

Denature

The process by which a protein loses its three-dimensional structure due to external stress or compounds, affecting its functionality.

Enzyme

Biological molecules that significantly speed up the rate of virtually all of the chemical reactions that take place within cells.

Nonpolar Covalent Bond

A type of chemical bond where two atoms share a pair of electrons equally, usually between identical or similar atoms.