Let F(x,y)=<>. Compute the flux oint F * nds of F across a simple closed curve that is the boundary of the half-disk given by x^2+y^2<=7, y>= 0 using the vector form Green's Theorem. oint F * nds = ?

Let F(x,y)=<<xy^2+8x, x^2y-8y>>. Compute the flux oint F * nds of F across a simple closed curve that is the boundary of the half-disk given by x^2+y^2<=7, y>= 0 using the vector form Green's Theorem. oint F * nds = ?

Question
Let \(\displaystyle{F}{\left({x},{y}\right)}={\left\langle{x}{y}^{{2}}+{8}{x},{x}^{{2}}{y}-{8}{y}\right\rangle}\). Compute the flux \(\displaystyle\oint{F}\cdot{n}{d}{s}\) of F across a simple closed curve that is the boundary of the half-disk given by \(\displaystyle{x}^{{2}}+{y}^{{2}}\le{7},{y}\ge{0}\) using the vector form Green's Theorem.
\(\displaystyle\oint{F}\cdot{n}{d}{s}=\) ?

Answers (1)

2020-12-18
Step 1
Consider the equation
Step 2
\(\displaystyle{F}{\left({x},{y}\right)}={\left({x}{y}^{{2}}+{8}{x},{x}^{{2}}{y}-{8}{y}\right)}\)
\(\displaystyle\oint{F}.{n}{d}{s}==\int\int\nabla\times{D}{d}{A}\)
\(\displaystyle\oint{F}.{n}{d}{s}=\int\int_{{S}}{\left({8}{x}{\left.{d}{y}\right.}+{y}\right)}={3}\pi\)
\(\displaystyle\int{\left({8}{x},{y}\right)}.{\left({x},{y}\right)}{d}{s}=\int{8}{x}{\left.{d}{y}\right.}+{y}{\left(-{\left.{d}{x}\right.}\right)}\)
\(\displaystyle{x}={\cos{{0}}}{y}={\sin{{0}}}{d}{s}={r}{d}{0}\Rightarrow{d}{s}={d}{0}\) because the radius is 1
\(\displaystyle{\left.{d}{x}\right.}=-{y}{d}{0}{\left.{d}{y}\right.}={\cos{{0}}}{d}{0}\)
\(\displaystyle\oint{\left(-{y},{8}{x}\right)}.{\left.{d}{x}\right.}\)
\(\displaystyle\int\int_{{S}}\frac{{\partial{N}}}{{\partial{x}}}-\frac{{\partial{M}}}{{\partial{y}}}={3}\)
(the area) \(\displaystyle={3}{\left(\pi{\left({1}\right)}^{{2}}\right)}={3}\pi\)
0

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