Determine Whether or Not $\mathbf { F } ( x , y ) = \left( 3 x ^ { 2 } - 4 y \right) \mathbf { i } + \left( 4 y ^ { 2 } - 2 x \right) \mathbf { j }$

question 36

Short Answer

Determine whether or not $\mathbf { F } ( x , y ) = \left( 3 x ^ { 2 } - 4 y \right) \mathbf { i } + \left( 4 y ^ { 2 } - 2 x \right) \mathbf { j }$ is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f$ .

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Definitions:

Arithmetic

A branch of mathematics dealing with the properties and manipulation of numbers, including operations such as addition, subtraction, multiplication, and division.

Selective Attrition

The process where certain kinds of participants drop out of a study over time, potentially skewing the results.

Demand Characteristics

Subtle cues or signals in an experimental setup that suggest to participants what behavior is expected or desired, potentially influencing the outcomes of the study.

Replicability

The ability of a study or experiment to be repeated or reproduced to verify results and confirm reliability and validity of findings.