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

question 51

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:

Antisocial Behavior

Actions that harm or lack consideration for the well-being of others; often violating social norms.

Interpersonal Relationships

Emotional, social connections or associations between two or more people, often characterized by varying degrees of intimacy and sharing.

Culturally Undesirable

Traits or behaviors that are rejected or looked down upon by a particular culture or society.

Violence

Aggression that has as its goal extreme physical harm, such as injury or death.