
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 27th, 2013, 11:06 AM  #1 
Senior Member Joined: Nov 2013 Posts: 137 Thanks: 1  Fundamental theorem of calculus for surface integrals?
Hellow! A simple question: if exist the fundamental theorem of calculus for line integrals not should exist too a fundamental theorem of calculus for surface integrals? I was searching about in google but I found nothing... What do you think? Such theorem make sense? 
December 27th, 2013, 04:15 PM  #2 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Fundamental theorem of calculus for surface integrals?
That may be because it is incorporated into a much more general theorem the generalized Stokes' theorem: if is a differential form defined on a differentiable manifold then (http://en.wikipedia.org/wiki/Stokes'_theorem). That is, the integral of the differential form, over region is the antiderivative, integrated (i.e. evaluated) over the boundary of the region . The reason why we have specific expressions for an integral on the real number line or on a path in two or three dimensions is that the boundary is just the two endpoints and in "integrating" on that boundary is just evaluating at the two endpoints and subtracting. In the case that is a two dimensional surface in three dimension, is the closed path boundary of the surface. 
December 27th, 2013, 07:14 PM  #3 
Senior Member Joined: Nov 2013 Posts: 137 Thanks: 1  Re: Fundamental theorem of calculus for surface integrals?
But, anyway, exist a geometric explanation for the fundamental theorem of calculus for surface integral, a geometric explanation analogous to t.f.c. for line integral?


Tags 
calculus, fundamental, integrals, surface, theorem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Fundamental theorem of calculus  layd33foxx  Calculus  3  December 12th, 2011 07:32 PM 
fundamental theorem of calculus  riotsandravess  Calculus  3  November 25th, 2010 12:44 PM 
Fundamental Theorem of Calculus  Aurica  Calculus  1  June 14th, 2009 08:04 AM 
Fundamental Theorem of Calculus  Aurica  Calculus  1  June 10th, 2009 05:39 PM 
Fundamental Theorem of Calculus  mrguitar  Calculus  3  December 9th, 2007 01:22 PM 