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4: Integral Theorems

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  • 4.1: Gradient, Divergence and Curl
    “Gradient, divergence and curl”, commonly called “grad, div and curl”, refer to a very widely used family of differential operators and related notations that we'll get to shortly. We will later see that each has a “physical” significance. But even if they were only shorthand, they would be worth using.
  • 4.2: The Divergence Theorem
    The rest of this chapter concerns three theorems: the divergence theorem, Green's theorem and Stokes' theorem. Superficially, they look quite different from each other. But, in fact, they are all very closely related and all three are generalizations of the fundamental theorem of calculus
  • 4.3: Green's Theorem
    Our next variant of the fundamental theorem of calculus is Green's 1 theorem, which relates an integral, of a derivative of a (vector-valued) function, over a region in the xy-plane, with an integral of the function over the curve bounding the region. First we need to define some properties of curves.
  • 4.4: Stokes' Theorem
    Our last variant of the fundamental theorem of calculus is Stokes' 1 theorem, which is like Green's theorem, but in three dimensions. It relates an integral over a finite surface in R3 with an integral over the curve bounding the surface.
  • 4.5: Optional — Which Vector Fields Obey ∇ × F = 0
    We already know that if a vector field F passes the screening test ×F=0 on all of R2 or R3, then there is a function φ with F=φ.
  • 4.6: Really Optional — More Interpretation of Div and Curl
    We are now going to determine, in much more detail than before, what the divergence and curl of a vector field tells us about the flow of that vector field.
  • 4.7: Optional — A Generalized Stokes' Theorem
    As we have seen, the fundamental theorem of calculus, the divergence theorem, Greens' theorem and Stokes' theorem share a number of common features. There is in fact a single framework which encompasses and generalizes all of them, and there is a single theorem of which they are all special cases.

Thumbnail: Diagram of an arbitrary volume partitioned into several parts, illustrating that the flux out of the original volume is equal to the sum of the flux out of the component volumes. (CC0; Chetvorno via Wikipedia)


This page titled 4: Integral Theorems is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform.

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