Use the method of this example to calculate f · dr,c wheref(x, y) = 2xyi (y2 − x2)j(x2 y2)2 and c is any positively oriented simple closed curve that encloses the origin. f · dr

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tanvigupta426 tanvigupta426 · Answer 1 · 2022-08-09T14:59:33+02:00

The area around the given curve according to the green theorem is [tex]F. dr = 0[/tex].

According to the statement

we have to find the area enclosed by the simple closed curve that encloses the origin.

So, We know that the

The given equation is

[tex]f(x,y) = \frac{2xyi + (y^{2} - x^{2} ) j}{(x^{2} + y^{2} )^{2} }[/tex]

and

If function is in form of,

[tex]F = Pi + Qj[/tex]

and C is any positively oriented simple closed curve that encloses the origin.

Then,by use of Green's theorem

Do the partial differentiation of the given function

Then

[tex]\frac{dQ}{dx} = \frac{2x^{3} - 6xy^{2}}{(x^{2} + y^{2} )^{3}}[/tex]

and

[tex]\frac{dP}{dy} = \frac{2x^{3} - 6xy^{2}}{(x^{2} + y^{2} )^{3}}[/tex]

On substitution in Green's theorem,

We get the value

[tex]F. dr = 0[/tex]

From this it is clear that the area around the given curve is zero.

So, The area around the given curve according to the green theorem is [tex]F. dr = 0[/tex].

Learn more about Green theorem here

https://brainly.com/question/23265902

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