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Physics 325 Lecture 3

# 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 ( ) A A t = G G . Then (3.1) ( ) ( ) ( ) ( ) 2 2 2 2 1 1 1 1 ˆ ˆ ˆ t t t t x y z t t 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 ˆ ˆ , , , , , , , , x y z V V 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 2 2 2 1 1 1 x y z V x y z 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 n i = ˆ ˆ x da ida idydz = = G ( ) , , A x y z G onto the normal of the surface as ( ) ˆ x x y y z z S S S A da A nda A da A da A da = = + + G G G i i (3.3)

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