Use Stokes' theorem to evaluate the line integral oint_C F*dr where A = -yi + xj and C is the boundary of the ellipse x^2/a^2+y^2/b^2=1, z = 0.

Use Stokes' theorem to evaluate the line integral oint_C F*dr where A = -yi + xj and C is the boundary of the ellipse x^2/a^2+y^2/b^2=1, z = 0.

Question
Use Stokes' theorem to evaluate the line integral \(\displaystyle\oint_{{C}}{F}\cdot{d}{r}\) where A = -yi + xj and C is the boundary of the ellipse \(\displaystyle\frac{{x}^{{2}}}{{a}^{{2}}}+\frac{{y}^{{2}}}{{b}^{{2}}}={1},{z}={0}\).

Answers (1)

2020-10-29
Step 1
Stoke's theorem given by
\(\displaystyle\int_{{C}}{F}.{d}{r}=\int\int_{{S}}{c}{u}{r}{l}{A}\times\hat{{{n}}}{d}{s}\)
According to the question,
A=−y i+x j
\(\displaystyle{C}{u}{r}{l}{A}={\left[\begin{array}{ccc} \hat{{{i}}}&\hat{{{j}}}&\hat{{{k}}}\\\frac{\partial}{{\partial{x}}}&\frac{\partial}{{\partial{y}}}&\frac{\partial}{{\partial{z}}}\\-{y}&{x}&{0}\end{array}\right]}\)
Step 2
Boundary condition of the curve C is,
\(\displaystyle\frac{{x}^{{2}}}{{a}^{{2}}}+\frac{{y}^{{2}}}{{b}^{{2}}}={1},{z}={0}\)
\(\displaystyle\hat{{{n}}}=\hat{{{k}}}\)
therefore,
\(\displaystyle{c}{u}{r}{l}{A}.{n}={2}\hat{{{k}}}.\hat{{{k}}}={2},{\left\langle.\hat{{{k}}}.\hat{{{k}}}={1}\right)}\)
Stoke's theorem given by
\(\displaystyle\int_{{C}}{F}.{d}{r}=\int\int_{{S}}{c}{u}{r}{l}{A}\times\hat{{{n}}}{d}{s}\)
\(\displaystyle=\int\int_{{S}}{2}{d}{s}\)
\(\displaystyle={2}{\left(\pi{a}{b}\right)}={2}\pi{a}{b}\)
0

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