Table of Contents 1. Line Integrals 2. Path Independence of Line Integrals 3. Double Integrals 4. Green’s Theorem in the Plane 5. Surfaces for Surface Integrals 6. Surface Integrals 7. Triple Integrals. Divergence Theorem of Gauss 8. Applications of Divergence Theorem 9. Stoke’s Theorem Appendix A-1 Equation of Continuity A-2 Equation of Conservation
Vector Integral Calculus 1. Line Integrals Vector Integral Calculus 1.Line Integrals Let Cbe a curve, there are several types of line integrals. Definition:Line IntegralLet ( ,, )fx y zbe a continuous function defined at all point of C. The line integral of ( ,, )fx y zover the curve Cis defined as 01( ,, )lim()knkkCskf x y z dsf Pswhere we partition the curve into sub-arcs with kP(any point in the k-th sub-arc) and ks(length of the k-th sub-arc). Cis called the path of integration that initial point A(initial point) and B(terminal point). Cis smooth curve if Chas unique tangent at each of points varies continuously. The direction from Ato Bis called the positivedirection on C(counterclockwise). Definition:Line IntegralLet 1(,,)iiiF xyz, 2(,,)iiiFxyzand 3(,,)iiiFxyzbe continuous differentiable functions defined on C. The line integral is 12312301lim(,,)(,,)(,,)kCnkkkkkkkkkkkkskF dxF dyF dzF xyzxFxyzyFxyzzwhere we partition the curve Cwith (,,)kkkxyz(any point in the k-th sub-arc) and , , kkkxyz(the usual difference in ,,x y z) . How to compute line integral To integrate a continuous function ( ,, )fx y zover a curve C: 1.Parameterize the curve C: ( ) ( ) ( ) ( )tx ty tz trijk2.Change variables (from sto tor from ( ,, )x y zto t) ( ,, )( ( ),( ), ( )) |( ) |bCafx y z dsfx ty tz ttdtr
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Vector Integral Calculus 1. Line Integrals Recall: The arc length of a curve Cfrom point Ato ( ,, )x y zis given by222///bbaasdsdxdtdydtdzdtdt1.1.Vector Form of Line Integrals Definition:Vector Line IntegralLet Cbe a curve given by a position vector ( ) [ ( ), ( ), ( )] ( )( )( )tx ty tz tx ty tz trijkand let ( ,, ) ( )x y zFF rbe a vector field defined on C. We define the line integral a( )( ( ))bCddtdtdtrF rrF ras follows ||01( )lim(,,)niiiiCidxyzrF rrFrwhere 1( ) ()iiittrrror iiiixyz rijk. ※work done by force Fin a displacement along C(work integral) ▶Closed Path―Aand Bmay coinside. ▶Piecewise Continuous―it consist of finitely many smooth curves. ▶Smooth curveC―Chas a unique continuous tangent or ( )tris differentiable and '( )/tddtrris continuous onC