Let V be a region in 3complying with the hypotheses of the divergence theorem, and denote by S its boundary surface. Let also φ: →  be a scalar function, and c an arbitrary constant vector. By applying the divergence theorem to the vector field φc (1) show that: (∫∫∫v ▼φdV - ∫∫s φndS).c = 0 with the understanding that the integral of a vector is the vector of the integrals of the components. (2) Use the above result to deduce carefully that: ∫∫∫v ▼φdV = ∫∫s φndS.

                                                        Real Analysis
                                                    Divergence Theorem

              
           Let V be a region in 3complying with the hypotheses of the divergence theorem,
           and denote by S its boundary surface. Let also φ: →  be a scalar function, and c an arbitrary constant vector.
                    
           By applying the divergence theorem to the vector field φc  
              (1) show that:

                                (∫∫∫v ▼φdV - ∫∫s φndS).c = 0    
                    with the understanding that the integral of a vector is the vector of the integrals of the components.

             (2) Use the above result to deduce carefully that:

                              ∫∫∫v ▼φdV = ∫∫s φndS.  
                                                                
                            See the attached file.

Attached file(s):

divergene1.doc  View File