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Ch_04 - Vector II - Summary

# Ch_04 - Vector II - Summary - Chapter 4 Vector Integral...

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Chapter 4 Vector Integral Calculus

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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 C be a curve, there are several types of line integrals. Definition: Line Integral Let ( , , ) f x y z be a continuous function defined at all point of C . The line integral of ( , , ) f x y z over the curve C is defined as 0 1 ( , , ) lim ( ) k n k k C s k f x y z ds f P s where we partition the curve into sub-arcs with k P (any point in the k -th sub-arc) and k s (length of the k -th sub-arc). C is called the path of integration that initial point A (initial point) and B (terminal point). C is smooth curve if C has unique tangent at each of points varies continuously. The direction from A to B is called the positive direction on C (counterclockwise). Definition: Line Integral Let 1 ( , , ) i i i F x y z , 2 ( , , ) i i i F x y z and 3 ( , , ) i i i F x y z be continuous differentiable functions defined on C . The line integral is 1 2 3 1 2 3 0 1 lim ( , , ) ( , , ) ( , , ) k C n k k k k k k k k k k k k s k F dx F dy F dz F x y z x F x y z y F x y z z where we partition the curve C with ( , , ) k k k x y z (any point in the k -th sub-arc) and , , k k k x y z (the usual difference in , , x y z ) . How to compute line integral To integrate a continuous function ( , , ) f x y z over a curve C : 1. Parameterize the curve C : ( ) ( ) ( ) ( ) t x t y t z t r i j k 2. Change variables (from s to t or from ( , , ) x y z to t ) ( , , ) ( ( ), ( ), ( )) | ( ) | b C a f x y z ds f x t y t z t t dt r

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Vector Integral Calculus 1. Line Integrals Recall: The arc length of a curve C from point A to ( , , ) x y z is given by 2 2 2 / / / b b a a s ds dx dt dy dt dz dt dt 1.1. Vector Form of Line Integrals Definition: Vector Line Integral Let C be a curve given by a position vector ( ) [ ( ), ( ), ( )] ( ) ( ) ( ) t x t y t z t x t y t z t r i j k and let ( , , ) ( ) x y z F F r be a vector field defined on C . We define the line integral a ( ) ( ( )) b C d d t dt dt r F r r F r as follows | | 0 1 ( ) lim ( , , ) n i i i i C i d x y z r F r r F r where 1 ( ) ( ) i i i t t r r r or i i i i x y z       r i j k . work done by force F in a displacement along C (work integral) Closed Path A and B may coinside. Piecewise Continuous it consist of finitely many smooth curves. Smooth curve C C has a unique continuous tangent or ( ) t r is differentiable and '( ) / t d dt r r is continuous on C