Determine Whether $\mathbf { F }$ Is Conservative $f$ Such That $\mathbf { F } = \nabla f$

question 35

Short Answer

Determine whether $\mathbf { F }$ is conservative. If so, find a function $f$ such that $\mathbf { F } = \nabla f$ .
$$\mathbf { F } ( x , y , z ) = 4 x ^ { 3 } y ^ { 2 } z ^ { 3 } \mathbf { i } + 2 x ^ { 4 } y z ^ { 3 } \mathbf { j } + 3 x ^ { 4 } y ^ { 2 } z ^ { 2 } \mathbf { k }$$

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Definitions:

Capacitive Reactances

The resistance a capacitor offers to alternating current, depending on the frequency of the current and the capacitance.

Series RC

A circuit configuration where a resistor and a capacitor are connected sequentially in a single path for current to flow.

Capacitive Circuit

A type of electrical circuit that primarily contains capacitors among other components, which stores and releases electrical energy.

Electric Degrees

A unit of measurement used in electrical engineering to describe phase angle, where 360 electric degrees correspond to one full cycle of a sine wave.