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The value of the integral 3ydx+2xdy using Green's theorem be - xy
The value of [tex]\int\limits_c 3ydx+2xdy[/tex] be -xy
What is Green's theorem?
Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C.
If M and N are functions of (x, y) defined on an open region containing D and having continuous partial derivatives there, then
[tex]\int\limits_c Mdx+Ndy[/tex] = [tex]\int\int\〖(N_{x}-M_{y}) \;dxdy[/tex]
Using green's theorem, we have
[tex]\int\limits_c Mdx+Ndy[/tex] = [tex]\int\int\〖(N_{x}-M_{y}) \;dxdy[/tex] ............................... (1)
Here [tex]N_{x}[/tex] = differentiation of function N w.r.t. x
[tex]M_{y}[/tex]= differentiation of function M w.r.t. y
Given function is: 3ydx + 2xdy
On comparing with equation (1), we get
M = 3y, N = 2x
Now, [tex]N_{x}[/tex] = [tex]\Luge\frac{dN}{dx}[/tex]
= [tex]\frac{d}{dx} (2x)[/tex]
= 2
and, [tex]M_{y}[/tex] = [tex]\Huge\frac{dM}{dy}[/tex]
= [tex]\frac{d}{dy} (3y)[/tex]
= 3
Now using Green's theorem
= [tex]\int\int\〖(2 -3) dx dy[/tex]
= [tex]\int\int\ -dxdy[/tex]
= [tex]-\int\ x dy[/tex]
=[tex]-xy[/tex]
The value of [tex]\int\limits_c 3ydx+2xdy[/tex] be -xy.
Learn more about Green's theorem here:
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