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

question 46

Essay

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

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Definitions:

Amoeboid Cells

Cells that move or feed by forming extensions of their cytoplasm, known as pseudopodia.

Rhizarians

A diverse supergroup of amoeboid cells that often have hard outer shells, called tests, through which cytoplasmic projections extend; include forams, actinopods, and certain shell-less amoebas.

Unikonts

A group of eukaryotic organisms that include amoebas, most animals, and fungi, characterized by a single emergent flagellum.

Evolutionary Significance

The impact or importance of evolutionary changes and adaptations in the history and development of species, populations, or genes.