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) \(1843.2 \pi\)

---

Definitions:

Respondent Behavior

Behavior that occurs as a direct response to a specific stimulus, often reflexive or involuntary in nature.

Classical Conditioning

A learning process in which a biologically potent stimulus is paired with a previously neutral stimulus to elicit a conditioned response.

Negative Reinforcement

A behavioral principle where the removal or avoidance of an unpleasant stimulus increases the likelihood of a behavior being repeated.

Instinctive Drift

The tendency of an animal to revert to instinctive behaviors that interfere with learned behaviors from operant conditioning.