Physics 325 Lecture 3 - Physics 325 Lecture 3 Integration...

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Physics 325 Lecture 3 Integration of Vectors Let be a vector function of t : A G ( ) AA t = G G . Then (3.1) () () () () 22 2 2 11 1 1 ˆ ˆˆ tt t t xyz t t A t dt i A t dt j A t dt k A t dt =++ ∫∫ ∫ ∫ G The vector integral is done by performing three separate ordinary integrations. Similarly, if the function depends on x,y,z , the integration over volume becomes () () ( ) ( ) ˆ ,, xy z VV V V A x y z dV i A x y z dV j A x y z dV k A x y z dV =+ + ∫∫∫ ∫ G (3.2) where 222 111 V dV dx dy dz = ∫∫∫ Similarly, the symbol denotes a double integral over a surface S . S Surface Integrals: Let da be an element of area of a surface S , and be the unit normal vector to S at da : ˆ n ˆ da da n = G da The vector is associated with the element of surface area da , and points in a direction that is outward from and normal to the surface. “Outward” is well defined for a closed surface. For an open surface, we will adopt the right-hand rule to define the “outward” direction. da G If , then . We can now define the integral over a surface of the projection of a vector function ˆ ˆ ni = x da ida idydz == G ( ) Axyz G onto the normal of the surface as ( ) ˆ xx yy zz SS S A da A nda A da A da A da ==+ + ∫∫ ∫ GG G ii (3.3)
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Line Integrals: Let L be a path from point P to point Q, and dl G be an incremental element of that path: The integral of a vector function A G along the path from P to Q is given by the integral of the projection of along the path A G dl G P Q ( ) xyz PQ PQ A dl A dx A dy A dz =+ + ∫∫ G G i (3.4) We now state, without proof, two important theorems: Gauss’s Theorm: SV Ada AdV =∇ G G G G ii (3.5) This relates the integral of a vector function over a closed surface to the volume integral of the divergence of the vector function (Equation (2.8)) over the volume enclosed by the surface.
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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Physics 325 Lecture 3 - Physics 325 Lecture 3 Integration...

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