Today in Physics 217: vector integrals
       Three useful generalizations 
       of the fundamental theorem                       vld
                                                   Γ=∫ ⋅
       of calculus:                                   v
        ‰Gradient theorem
        ‰Gauss’ divergence 
          theorem
        ‰Stokes’ curl theorem
       9 September 2002         Physics 217, Fall 2002              1
                              Integral vector calculus
         Fundamental theorem of calculus for a function of one 
         variable:
                                b df  x
                                     ()
                                          dx =−f b      f  a
                               ∫                  () ()
                                a   dx
         In vector calculus, there are three different kinds of 
         derivatives – gradient, divergence and curl – so there are 
         three different analogues of the fundamental theorem of 
         calculus: 
          ‰the                          :
                  gradient theorem
                                  ∇ ⋅= −
                                         lba
                                    Td T T
                                ∫              () ()
                                C
             where the integral is taken along the curve  , and a and b
                                                                    C
             are the position vectors of the endpoints of  .
                                                                    C
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                 Integral vector calculus (continued)
        ‰Stokes’ theorem, for curls:
                                    dd
                                 va vl
                              ∇×=⋅
                             ()
                           ∫∫
                                        v
                           SC
          where the integral on the left is carried out over a surface 
            , and that on the right is carried out all the way around 
          S
          the curve    that bounds 
                    C              S. 
        ‰And (Gauss’)                        : 
                         divergence theorem
                                    dd
                               ∇⋅=vvτ       ⋅a
                              ()
                            ∫∫
                                         v
                            VS
          where the integral on the left is carried out over a volume 
            , and that on the right over the surface  that bounds  .
          V                                         S             V
       Illustrating these theorems one by one…
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                            Gradient theorem
                                 lba
                            ∇Td⋅=T −T
                          ∫           () ()
                          C
        ‰The left-hand side is a line integral. It is evaluated by 
          choosing a specific path from a to b. 
        ‰The theorem ensures that the result is independent of the 
          path chosen. (So choose one that makes the integral 
          easy…) This is not true of arbitrary vector functions: only 
          gradients have this property. 
        ‰The line integral of the gradient of  around a closed loop 
                                              T
          is zero:                  ()
                               laa0
                          ∇Td⋅=T −T =
                        v                  ()
                        ∫
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