Vector calculusDate: Fri, 18 Nov 1994 08:06:29 -0800 (PST) From: Uma Chandavarkar Hello! I am a senior at Monta Vista High School, and am currently enrolled in Calculus. I have been studying vector calculus, and I was interested in finding more about the principles of the divergence theorem and Stokes' theorem, which relate to flux and curl, respectively. Thank you. - Uma Chandavarkar Date: Fri, 18 Nov 1994 11:52:04 -0500 (EST) From: Dr. Ken Subject: Re: Vector Calculus Hello Uma! Ah, Stokes' theorem. A beautiful theorem. I never could figure out, though, why they don't just call all those kinds of theorems "The Fundamental Theorem of Calculus" and leave it at that, rather than giving them all kinds of different names like Green's theorem, Gauss's theorem, and whatever all the rest are. I guess it's the engineers and the physicists. Anyway, if you've seen Stokes' theorem, you've probably been told that it's the n-dimensional analogue of the Fundamental theorem of Calculus you saw in regular one-dimensional calculus. The integral of a differential form over the boundary of a closed region is equal to the integral of the differential of that form over the region itself. Nice stuff. In one dimensional work, you do it like so: you've got an integral of a one-form F (is this terminology you've seen before?) over an interval in R1, and that's equal to the integral (i.e. the sum) of a form K which has F as its differential over the boundary of the interval (i.e. the endpoints of the interval). In practice, you just say K is an antiderivative of F. This is a pretty deep topic, and Multivariable Calculus is essentially the study of trying to understand it. So if you're taking the course, you're in the right place. -Ken "Dr." Math |
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