数学题在线解答

问题:Compute the outward flux of the vector field F= (5xy, 5y^2) for the region enclosed by the curves y=x^2 and y=x in the first quadrant, without parameterizing the curves (i.e. use the flux version of Green’s theorem).

解答

Let A be the region enclosed by the curves. Let C be the boundary of the region, oriented counterclockwise. By Green’s theorem, we have

\begin{aligned}
I &= \oint_C F \cdot n {d}s\\
&= \oint_C 5xy\mathrm{d}y – 5y^2\mathrm{d}x\\
&= \iint_A(5y+10y)\mathrm{d}\sigma\\
&= 15\iint_Ay\mathrm{d}\sigma\\
&= 15\int_0^1\mathrm{d}x\int_{x^2}^x y \mathrm{d}y\\
&= 15\int_0^1 (x^2/2-x^4/2) \mathrm{d}x\\
&= 15(1/6-1/10)\\
&= 1
\end{aligned}

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Moore 2020-12-15 05:08:43
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