The Following Table Gives Values of a Function F(x, Y)

question 4

Essay

The following table gives values of a function f(x, y).The table reflects the properties of the function, which is differentiable and defined for all (x, y). $\begin{array}{c}x\\y\begin{array}{|r|r|r|c|c|c|c|c|c|c|c|}\hline &1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\hline 1 & 7 & 8 & 4 & 3 & 8 & 2 & 1 & 1 & 5 & 9 \\\hline 2 & 5 & 9 & 7 & 11 & 7 & 3 & 1 & 4 & 2 & 10 \\\hline 3 & 6 & 8 & 14 & 11 & 10 & 5 & 13 & 13 & 12 & 14 \\\hline 4 & 11 & 15 & 20 & 22 & 25 & 24 & 21 & 21 & 15 & 12 \\\hline 5 & 17 & 25 & 30 & 31 & 32 & 35 & 37 & 40 & 35 & 32 \\\hline 6 & 25 & 30 & 34 & 30 & 29 & 26 & 15 & 14 & 12 & 9 \\\hline 7 & 39 & 42 & 51 & 55 & 50 & 49 & 47 & 45 & 35 & 36 \\\hline 8 & 26 & 21 & 19 & 24 & 28 & 27 & 30 & 33 & 45 & 39 \\\hline 9 & 49 & 50 & 55 & 62 & 69 & 71 & 60 & 54 & 49 & 47 \\\hline 10 & 65 & 70 & 64 & 6 & 63 & 49 & 42 & 41 & 40 & 38 \\\hline\end{array}\end{array}$ Let $\vec{F}=\nabla f$ and $\vec { G } = 6 \overline { \bar { F } } .$ Find $\int _ { C } \vec { G } \cdot d \vec { r }$ if C consists of line segments connecting (1,1), (1,10), (5,6), and (9,8)in that order.Explain your reasoning.

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Definitions:

1-Methylcyclohexane-1,4-diol

A compound featuring a cyclohexane ring with a methyl group and two hydroxyl groups positioned at the 1 and 4 positions.

Benzyl Alcohol

A type of alcohol with the formula C6H5CH2OH, characterized by a benzene ring attached to a CH2OH group.

Methyl Benzoate

An organic compound formed from the esterification of benzoic acid with methanol, characterized by a sweet smell and used in perfumery.

Aldehyde Treatment

A chemical process involving the reaction of an aldehyde group, often resulting in the introduction of new functional groups.