Show That F Is Conservative, and Find a Function F $\mathbf { F } = \nabla f$

question 105

Multiple Choice

Show that F is conservative, and find a function f such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 6 x y \mathbf { i } + \left( 3 x ^ { 2 } + 4 y z ^ { 2 } \right) \mathbf { j } + 4 y ^ { 2 } z \mathbf { k }$ ; $A ( 0,0,0 )$ and $B ( 2 , - 2,0 )$

• A) -113
• B) -24
• C) -131
• D) -178

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Definitions:

Growth Hormone

A hormone secreted by the pituitary gland that stimulates growth, cell reproduction, and regeneration in humans and other animals.

Bone Growth Spurt

A rapid increase in bone length and density, typically occurring during puberty.

Sex Hormones

Hormones that are primarily involved in the development and regulation of sexual characteristics and reproductive functions.

Appositional Bone Growth

The process by which new bone tissue is added on the surface of existing bones, contributing to the increase in bone diameter.