Comments on Colley, Section 7.3 This section describes two 3 – dimensional generalizations of Green’s Theorem. One of them (Stokes’ Theorem) extends the usual form of Green’s Theorem involving the integral of F · ds, and the other (sometimes also called Green’s Theorem, but more often called the Divergence Theorem or Gauss’s Theorem or the Gauss – Ostrogradsky Theorem) extends the divergence form of Green’s Theorem involving the integral of F · d N. BACKGROUND REFERENCES. The divergence and curl of a vector field were defined and studied in Section 3.4 of the text; the latter section was covered in the first course of the multivariable calculus sequence (Mathematics 10A). The following document contains some additional information and online references: http://math.ucr.edu/~res/math10A/weblinks3.pdf Up to this point in the course, most of the emphasis has been upon the basic definitions of key concepts (various sorts of integrals, “nice” regions in the coordinate plane and 3 – space, potential functions, surface area, flux of a vector field, … ) and methods for computing various integrals and functions, including statements of several fundamentally important theorems like the Change of Variables Rule and Green’s Theorem. While these points are also central to the remaining sections of the course, there will also be an additional factor: We shall also focus on the logical derivations of some important formulas that play crucial roles in the applications of vector analysis to the sciences and engineering. Stokes’ Theorem This is result which contains the previously stated Green’s Theorem (involving the line integral of F · ds over a closed curve ΓΓΓΓ, or a collection of closed curves, bounding a region D) as a special case. The main difference is that the region D is replaced by a coherently oriented surface S. There is a discussion of coherent orientations at the end of the commentary for the previous section: http://math.ucr.edu/~res/math10B/comments0702.pdf The underlying ideas regarding coherent orientations are summarized in the following drawings. The first two pictures illustrate the relationship between the direction sense of ΓΓΓΓ and the orientation of S, while the third illustrates how orientations fit together for the parametrized pieces of a piecewise smooth surface.(Source: http://www.ittc.ku.edu/~jstiles/220/handouts/Stokes%20Theorem_1.png) (Source: http://en.wikipedia.org/wiki/File:Stokes%27_Theorem.svg)(Source: http://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Stokes.png/250px-Stokes.png) The third picture reflects the basic principle in the previously cited commentary for Section 7.2 of the text; namely, if one has a coherent orientation for S then the sums of the line integrals over the pieces of S will reduce to the line integral over the boundary curve(s). Yet another way of describing the relation between the orientation of S and the direction sense of the boundary curve ΓΓΓΓ is the following Right Hand Rule: If you right hand curls around the normal vector N in the directed sense of ΓΓΓΓ, then your thumb will be pointing in the direction of N. With the preceding conventions, we may state Stokes’ Theorem as follows: Let S be an oriented surface with unit coherent normal vector system N, and let ΓΓΓΓ be the positively oriented boundary of S. If F is a vector field with continuous first order partial derivatives which is defined in a region containing S, then we have the following identity:Frequently one sees the displayed equation in an equivalent form using notational conventions such as the ∇∇∇∇ operator; here is a typical example: There are also versions of Stokes’ Theorem for oriented surfaces whose boundaries consist of several curves; in these cases, the right hand side is replaced by a sum of line integrals over the various boundary curves, where each has the sense it inherits from the coherent orientation system for the surface. On the other hand, Stokes’ Theorem does not apply to the Möbius strip discussed in the commentary for Section 7.2; as indicated there, the sum of the line integrals over the parametrized pieces is never equal to the line integral along the boundary. No matter which orientation system one chooses for the parametrized pieces of the Möbius strip, the sum of the line integrals will always contain an extra term given by twice a line integral over a curve which is not contained in the boundary of the surface. A computational example. This is taken from the following online source: http://ltcconline.net/greenl/courses/202/vectorIntegration/stokesTheorem.htm Let S be the part of the plane z = 4 – x – 2y in the first octant, with upwardly pointing unit normal vector. Use Stokes' theorem to find where F(x, y, z) = (y, z,– xy).SOLUTION. We first notice that, without Stokes’ theorem, it would be necessary to parametrize three different line segments. Instead, we can evaluate the line integral by means of just one double integral. The first step is to find a suitable parametrization for the solid triangle, including the domain on which this parametrization is defined. The graph parametrization (x, y, 4 – x – 2y) will be fine for our purposes, and its domain of definition is the “shadow” that S casts upon the xy – plane, which is just the region in the first quadrant bounded by the coordinate axes and the line x + 2y = 4. We then have and N = i + 2j + k, so that Curl F · N = 1 + x + 2y - 1 = x + 2y. Therefore the line integral in the problem is given by and from this one sees that the value of the integral is 32/3. More conceptual implications of Stokes’ Theorem On a theoretical level¸ one important consequence of Stokes’ Theorem is its role in proving a fundamental result from Theorem 6.3; namely, a vector field F over a simply connected region ΩΩΩΩ in 3 – dimensional space is expressible the gradient ∇∇∇∇g of some function g on ΩΩΩΩ if and only if ∇∇∇∇ × F = 0. Here is a brief sketch of how one derives this fact. Let p be some fixed point in ΩΩΩΩ, for each x in the region pick some simple piecewise smooth path ΓΓΓΓ joining p to x, and set g( x ) equal to the line integral of F over ΓΓΓΓ. . . . In order to justify this definition, one needs to