Math277GreensTheorems

Math277GreensTheorems - Green's Theorem(s) In the...

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Green's Theorem(s) In the Plane--general conditions ì G is a simple, piecewise smooth, closed curve in the plane. [C is defined parametrically by < Ð>Ñß + Ÿ > Ÿ , Ð+Ñ o Ð,Ñß ß < < and the curve does not intersect itself anywhere else.] ì H is the plane region bounded by C. D contains the points inside and on C. ì + C is parameterized so that the region D is on the left as the parameter increases from to counterclockwise orientation) ,Þ Ð ì TÐBß CÑß UÐBß CÑ are defined and have continuous partial derivatives on an open region that contains D. [You can think of these functions individually or as the components of a vector field ( ) = ] J 3 4 5 Bß C TÐBß CÑ  UÐBß CÑ  ! Green's Theorem Tangential Component or Circulation-Curl Form). Ð ' ' ' G + + . .> Ð>Ñ , , J † < o J † J † < J † m< m . .> o ÒBÐ>Ñß CÐ>ÑÓ Ð>Ñ .> o Ð>Ñ .> o ' + , < w w < Ð m< m w w ' ' ' G G G .B .B .= .= . .C .C J † X .= o ÒTÐBß CÑ  UÐBß CÑ Ó .= o ÒTÐBß CÑ  UÐBß CÑ Ó . t dt t = = ' ' G w w ÒTÐBß CÑB Ð>Ñ  UÐBß CÑC Ð>ÑÓ .> T.B  U.C o G ' ' ' ' ' ' V V V Ð Ñ .E o -?<6 .E o Ðf ‚ .E `U `B `C `T J 5 JÑ 5 Notes: .= In the line integral, , is the scalar tangential component of the field at ' G J † X J † X a point on the curve and is the incremental distance along the curve. If is a .= J force field, the resultant integral represents done by the field in moving along the curve C. work If is a field, the resultant integral represents the curve C. (It J velocity flow along represents if C is a closed curve as it is for Green's Theoroem.) circulation ‡ Ð Ñ ÐBß CÑ Þ -?<6 ÐBß CÑÞ is a function of on R It is the component of It may `U `B `C `T 5 J also be shown to equal the "circulation density" at
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Math277GreensTheorems - Green's Theorem(s) In the...

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