18 . 024 – ESG Problem Set 12 Pramod
نویسنده
چکیده
Pramod N. Achar Spring 2000 In this problem set you will prove an analogue of the Fundamental Theorem of Calculus, Stokes’ Theorem, etc. To set up the context of the problem, let us define an operator div◦ on vector fields on R by div◦(P (x, y)i + Q(x, y)j) = ∂Q ∂x − ∂P ∂y . Let C denote a curve in R with endpoints a and b. Let S denote a surface whose boundary is the closed curve BdS. Let V denote a solid region in R with boundary BdV . The following table summarizes the state of our knowledge of integration theory in three or fewer dimensions. R R 2 R 3
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18 . 024 – ESG Notes 3
In this set of notes, we will develop the basic theory of tensors and differential forms; we will learn what it means to integrate a differential form; and we will state the generalized Stokes’ Theorem in terms of differential forms. Caveat lector: there are several closely related meanings of the word tensor. We begin introducing some convenient terminology for which we will not give precise d...
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