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MAT Integration Integration

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1. Line Integrals The line integral of the scalar field Ф over the curve c is denoted by Note : When the scalar field Ф is identically 1 gives the length of the curve. E.g. Evaluate the line integrals of The scalar field over the following lines. (2, 1, 0 ) (2, 1, 2 )

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Line integral of a vector field

Definition Line integral of a vector field We denote the integral of the vector field A over the curve c by E.g. Evaluate the integral of the unit tangent drawn to the unit circle.

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Flux – i%djh Suppose c is a simple closed curve in the space. The outward flux of the vector field A over c is denoted by , and defined as Here, s-arc length. normal tangent parallel to parallel to Outward unit normal is

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Let c ir, ixjD; jl%h msrsjid msg;g A ys i%djh

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Circulation – ixirKh Circulation is defined by

c ir, ixjD; jl%h msrsjid msg;g A ys ixirKh

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Green’s Theorem in 2-dimension – oaúudkfha .%skaf.a m%fushh

Suppose c is a simple closed curve in a plane, and S is the surface enclosed by c. The outward flux of a vector field A over c is same as the surface integral of div A over S. c ir, ixjD; jl%h msrsjid msg;g A ffoYsl lafIa;%fhys i%djh iy S msrsjid wmid A ys mDIaG wkql,h iudk fjs . since div So Green’s Theorem can be written as

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Circulation Flux form of Green’s Theorem.

.%skaf.a m%fushfha i%dj wdldrh Flux form of Green’s Theorem. Circulation .%skaf.a m%fushfha ixirK wdldrh Circulation form of Green’s Theorem.

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E.g. Solution : Evaluate . Here c ic the boundary of the triangle Let

Integral becomes

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E.g. Verify the flux form of the Green’s Theorem, by taking and

Flux form (i%dj wdldrh) of the Green’s Theorem is

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iu.ska i;Hdmkh lsrSu L.H.S.= jD;a;h mrdï;slj

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R.H.S.

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With Polar cdts jus mi yd ol=Kq mi iudk ksid .%skaf.a m%fushfha i%dj wdldrh i;Hdmkh fjs.

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