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

Short Answer

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 $\overline{\mathrm{H}}=7 \overline{\bar{F}} .$ Find $\int_{C} \bar{H} \cdot \overline{d r}$ if C is the circle of radius 2 centered at (4, 3).