Using the Divergence Theorem, Find the Outward Flux of F

question 24

Multiple Choice

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
- $\mathbf { F } = x \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathrm { i } + \mathrm { y } \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathrm { j } + \mathrm { z } \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathbf { k } ;$ D: the thick cylinder $5 \leq \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } \leq 6,3 \leq \mathrm { z } \leq 4$

• A) \(728 \pi\)
• B) \(\frac { 728 } { 3 } \pi\)
• C) \(\frac { 1456 } { 3 } \pi\)
• D) \(\frac { 364 } { 3 } \pi\)

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Definitions:

Zone of Proliferation

A region in growing tissues, such as bones, where cells are actively dividing.

Long Bones

Bones that are longer than they are wide, primarily found in the limbs and functioning to support body weight and enable movement.

Endochondral Ossification

The process by which bone tissue is formed from cartilage, a key mechanism in the growth of long bones in the body.

Intramembranous Ossification

A process of bone development in which bone forms directly from mesenchymal tissue without a cartilage precursor.