Find the Gradient Field of the Function $$f ( x , y , z ) = \left( \frac { x + y } { y + z } \right) ^ { 6 }$$

question 262

Multiple Choice

Find the gradient field of the function.
- $$f ( x , y , z ) = \left( \frac { x + y } { y + z } \right) ^ { 6 }$$

• A) \({\nabla} f = \frac { 6 ( x + y ) ^ { 5 } } { ( y + z ) ^ { 6 } } i + \frac { 6 ( x + y ) ^ { 5 } ( z - x ) } { ( y + z ) ^ { 7 } } j - 6 \frac { ( x + y ) ^ { 6 } } { ( y + z ) ^ { 7 } } \mathbf { k }\)
• B) \({\nabla} f = 6 \left( \frac { x + y } { y + z } \right) ^ { 5 } i + \frac { 6 ( x + y ) ^ { 5 } ( z + x ) } { ( y + z ) ^ { 7 } } j + 6 \frac { ( x + y ) ^ { 6 } } { ( y + z ) ^ { 7 } } k\)
• C)\({\nabla} f = 6 \left( \frac { x + y } { y + z } \right) ^ { 5 } i + \frac { 6 ( x + y ) ^ { 5 } ( z - x ) } { ( y + z ) ^ { 7 } } j + 6 \frac { ( x + y ) ^ { 6 } } { ( y + z ) ^ { 7 } } \mathbf { k }\)
• D)\({\nabla} f = 6 \left( \frac { x + y } { y + z } \right) ^ { 5 } i + \frac { 6 ( x + y ) ^ { 5 } ( z + x ) } { ( y + z ) ^ { 7 } } j - 6 \frac { ( x + y ) ^ { 6 } } { ( y + z ) ^ { 7 } } k\)

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Definitions:

Proprioception

The awareness of the positioning and coordination of one's body parts, along with the amount of force being applied during physical activity.

Sensory Receptors

Specialized cells or nerve endings that respond to changes in the environment by generating electrical signals.

Meissner Corpuscles

Nerve endings in the skin that are specialized to detect light touch sensitivity.

Pacinian Corpuscle

Named for Italian anatomist Filippo Pacini (1812–1883). See lamellated corpuscle.