Subject: Calculus

Greens Theorem

Calculus.GreensTheorem History

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July 03, 2011 by matthew_suan -
July 03, 2011 by matthew_suan -
Added line 28:
'''NEXT TOPIC''': [[Divergence theorem]]
March 20, 2011 by matthew_suan -
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'''Solution'''. We can actually solve this by having three different integrals of the functions from (0,0) to (1,0) , from (1,0) to (0,1), and from (0,1) to (0,0). Here, we would like to show a different approach using the Greenís Theorem.
We let {$P(x,y)= x^4 $} and {$Q(x,y)= xy$}. By Greenís theorem, we have
March 20, 2011 by matthew_suan -
March 20, 2011 by matthew_suan -
Changed lines 13-23 from:
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!!!Example#1

Evaluate

{$$\int_{C} x^4 dx+xydy$$}

where {$C$} is the triangular curve consisting of the line segments from (0,0) to (1,0) , from (1,0) to (0,1), and from (0,1) to (0,0) .

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March 20, 2011 by matthew_suan -
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Take for example the figure shown below. {$C$} here corresponds to an oriented close path or a path where the starting point is the same as the endpoint. {$D$} here is the interior region of closed path {$C$}. Greenís Theorem transforms a line integral at a closed curve {$C$} and a double integral over the plane region {$D$} bounded by {$C$}.
to:
Take for example the figure shown below. {$C$} here corresponds to an oriented close path or a path where the starting point is the same as the endpoint. {$D$} here is the interior region of closed path {$C$}. Greenís Theorem transforms a line integral at a closed curve {$C$} and a double integral over the plane region {$D$} bounded by {$C$}.

%center%Attach:green1.png

In stating Greenís Theorem, we use the convention that the positive orientation of a simple closed {$C$} refers to a single counterclockwise traversal of {$C$}. This means that if {$C$} is given by the vector function of {$r(t)$}, {$a \leq t \leq b$}, then the region {$D$} is always on the left as the point {$r(t)$} go across {$C$} as shown in the figure below.

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March 20, 2011 by matthew_suan -
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%cframe%{$\int_{C}Pdx+Qdy=\int_{D}\int \left (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right )dA$}
to:
%cframe%{$\int_{C}Pdx+Qdy=\int_{D}\int \left (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right )dA$}

!!Discussion
Take for example the figure shown below. {$C$} here corresponds to an oriented close path or a path where the starting point is the same as the endpoint. {$D$} here is the interior region of closed path {$C$}. Greenís Theorem transforms a line integral at a closed curve {$C$} and a double integral over the plane region {$D$} bounded by {$C$}.
March 20, 2011 by matthew_suan -
Added lines 1-4:
!!Green's Theorem
Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. If P and Q have continuous partial derivatives on an open region that contains D, then

%cframe%{$\int_{C}Pdx+Qdy=\int_{D}\int \left (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right )dA$}