Statistics Solutions

(Steps Shown) Here's the rectangle R with corners at -2,0,3,0,3, π and -2, π Use a 2 D integral to measure the net flow of the vector field \text Field

Question: Here's the rectangle \(\mathrm{R}\) with corners at \(\{-2,0\},\{3,0\},\{3, \pi\}\) and \(\{-2, \pi\}\)

Use a \(2 \mathrm{D}\) integral to measure the net flow of the vector field

\[\text { Field }[x, y]=\left\{x \operatorname{Cos}[y], e^{-x^{2} / 2}+y^{2}\right\}\]

across the boundary curve \(\mathrm{C}\) of this rectangle.

Say why you are happy to make this measurement by calculating your \(2 \mathrm{D}\) integral instead of making this measurement by calculating the path integral

\[\oint_{\mathrm{C}}-\left(\mathrm{e}^{-x^{2} / 2}+\mathrm{y}^{2}\right) d \mathrm{x}+\mathrm{x} \operatorname{Cos}[\mathrm{y}] d \mathrm{y}\]

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