Mathematics: Use green s theorem to evaluate the line integral along the given positively oriented curve. use green s theorem to evaluate the line integral c 5y3 dx ? 5x3 dy use green s theorem to evaluate the line integral c is the circle x2 + y2 = 4

It is not an exact representation of the atom, so it is not very useful....

Use green s theorem to evaluate the line integral

Ответ:

leilalony2660

03.01.2022

By Green's theorem,

$\displaystyle\oint_C(5y^3\,\mathrm dx\pm5x^3\,\mathrm dy)=\iint_D\left(\pm\frac\partial{\partial x}[5x^3]-\frac\partial{\partial y}[5y^3]\right)\,\mathrm dx\,\mathrm dy$

where

is the disk of radius 2 centered at the origin.

Differentiating and converting to polar coordinates, you have

$\displaystyle\iint_D\left(\pm15x^2-15y^2\right)\,\mathrm dx\,\mathrm dy=15\int_0^{2\pi}\int_0^2\left(\pm r^2\cos^2\theta-r^2\sin^2\theta\right)r\,\mathrm dr\,\mathrm d\theta=15\int_0^{2\pi}\int_0^2\left(\pm \cos^2\theta-\sin^2\theta\right)r^3\,\mathrm dr\,\mathrm d\theta$

which, depending on the sign concealed by the question mark, could be reduced to either

$\displaystyle15\int_0^{2\pi}\int_0^2r^3\cos2\theta\,\mathrm dr\,\mathrm d\theta=0$

if a plus, or

$\displaystyle-15\int_0^{2\pi}\int_0^2r^3\,\mathrm dr\,\mathrm d\theta=-8\pi$
Use green's theorem to evaluate the line integral along the given positively oriented curve. use gre

Use green's theorem to evaluate the line integral along the given positively oriented curve. use gre

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Use green's theorem to evaluate the line integral along the given positively oriented curve.
use green's theorem to evaluate the line integral
c
5y3 dx ? 5x3 dy
use green's theorem to evaluate the line integral
c is the circle x2 + y2 = 4

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Use green's theorem to evaluate the line integral along the given positively oriented curve. use green's theorem to evaluate the line integral c 5y3 dx ? 5x3 dy use green's theorem to evaluate the line integral c is the circle x2 + y2 = 4

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Use green's theorem to evaluate the line integral along the given positively oriented curve.
use green's theorem to evaluate the line integral
c
5y3 dx ? 5x3 dy
use green's theorem to evaluate the line integral
c is the circle x2 + y2 = 4