the folium of Descartes the folium of Descartes is an algebraic curve defined by the equationalgebraic curve When a = 1, parametrically, we can express part of the curve of looped shaped as

We have area

so, area

MAT iafgdalaiaf.a m%fushh

iafgdalaiaf.a m%fushh Stokes’s Theorem f uys mDIaG wkql,fha tall wNs,usnh i|yd Ok osYdj, c odrh osf.a jdudj¾: osYdjg.uka lrk úg iqr;a moaO;shla idok osYdjg fjs. S hkq ir, ixjD; odrhla iys; mDIaGhla hehs.ksuq. L et S be a surface and, c be the simple closed curved edge of S. ika;;sl wjl, ix.=Kl iys; A ffoYsl lafIa;%hla i|yd For a vector field A, with continuous partial derivatives fuys c hkq S m DIaGfha odrh fjs. c is the edge of the surface S.

Stokes’s Theorem Here, positive direction for the surface integral is the unit vector making the right handed system when we move in anticlockwise direction along the curve. fuys mDIaG wkql,fha tall wNs,usnh i|yd Ok osYdj, c odrh osf.a jdudj¾: osYdjg.uka lrk úg iqr;a moaO;shla idok osYdjg fjs. Eg: ffoYsl lafIa;%h iy mDIaGh ie,lSfuka iafgdalaiaf.a m%fushh i;Hdmkh lrkak.

Note: A line integral and a surface integral are connected by Stokes’s Theorem. Eg: Verify the Stokes’s Theorem with and the surface. C : Where.

Hence, Theorem is verified.

Verify the Stokes’s Theorem with and the surface. E.g.

Hence, Theorem is verified. E.g. Evaluate for The plane z = 2 Let

By Stokes theorem = o.

along the boundaries of the rectangle E.g. Evaluate Choose

o π 1

( 1, 0) ( -1, 0) ( 0, 1) ( 0, -1) E.g. Evaluate taken round the square C with vertices ( 1, 0),(-1, 0), ( 0, 1) and (0, -1)

( 1, 0) ( -1, 0) ( 0, 1) ( 0, -1)

Divergence Theorem Let S be a closed surface and V be the volume enclosed by S. Then outward flux of continuously differential Vector Field A over S is same as the volume integral of div A. i. e wmidrs;d m%fushh A hkq hus ixjD; mDIaGhla ;=< iy u; ika;;slj wjl,H ffoYsl flIa;%hla kus, tu mDIaGh msrsjid msg;g A ys i%djh, mDIaGfhka wdjD; jk m%foaYh msrsjid wmid A ys wkql,hg iudk fjs.

The Divergence Theorem connects a volume integral and an integral over a closed surface. Letting. A( plane surface) S o ( curve surface) Note : E.g. verify the Divergence Theorem for

For the curved surface S o Hence,

Further, gives Hence, the Divergence Theorem is verified. E.g. Evaluate, S is the closed surface bounded by the cone and the plane

Solution. X Y Z

* Evaluate. where S is the surface of the cube * Suppose S is the surface of the sphere of radius a centered at O, evaluate. * For any closed surface S, evaluate. Here, Divergence Theorem can not be applied. Why? Result If A.c = o for any constant arbitrary vector c, then A = 0.

1. If is a scalar field show that. Solution : Let be an arbitrary constant vector. Apply divergence Theorem to

. 2. For any closed surface s, show that Solution : Let be an arbitrary constant vector.

Since gives us Hence,.

E.g. If s is the surface of the sphere Show that Solution :

* Evaluate. where S is the surface of the cube X Y Z Here Apply divergence Theorem

Evaluate E.g. where c is the boundary of the surface in Oxy plane, enclosed by Ox axis and the semi-circle Flux form of Green’s Theorem..%skaf.a m%fushfha i%dj wdldrh

Now use polar coordinates.

Verify the circulation form of the Green’s Theorem in Oxy plane for and c is the curve and for E.g. Suppose that c is a simple closed curve Oxy plane not enclosing the origin O, evaluate for E.g.