Physics 217, Fall 2002 9 September 2002(c) University of Rochester 19 September 2002 Physics 217, Fall 2002 1Today in Physics 217: vector integralsThree useful generalizations of the fundamental theorem of calculus: Gradient theorem Gauss’ divergence theorem Stokes’ curl theoremdΓ= ⋅∫vlv9 September 2002 Physics 217, Fall 2002 2Integral vector calculusFundamental theorem of calculus for a function of one variable:In vector calculus, there are three different kinds of derivatives – gradient, divergence and curl – so there are three different analogues of the fundamental theorem of calculus:  the gradient theorem:where the integral is taken along the curve C, and a and bare the position vectors of the endpoints of C.()() ()badf xdxfbfadx=−∫()()CTd T T⋅= −∫lb a∇9 September 2002 Physics 217, Fall 2002 3Integral vector calculus (continued) Stokes’ theorem, for curls:where the integral on the left is carried out over a surface S, and that on the right is carried out all the way around the curve C that bounds S.  And (Gauss’) divergence theorem: where the integral on the left is carried out over a volume V, and that on the right over the surface S that bounds V.Illustrating these theorems one by one…()SCdd×=⋅∫∫va vlv∇()VSddτ⋅=⋅∫∫vvav∇Physics 217, Fall 2002 9 September 2002(c) University of Rochester 29 September 2002 Physics 217, Fall 2002 4Gradient theorem The left-hand side is a line integral. It is evaluated by choosing a specific path from a to b.  The theorem ensures that the result is independent of the path chosen. (So choose one that makes the integral easy…) This is not true of arbitrary vector functions: only gradients have this property.  The line integral of the gradient of T around a closed loop is zero:()()CTd T T⋅= −∫lb a∇()()0Td T T⋅=−=∫la av∇9 September 2002 Physics 217, Fall 2002 5Example illustration of the gradient theoremExample 1.6 in Griffiths: Verify thegradient theorem for this functionand these bounds,using both of the paths shown atright.1 212yx(2)(1)ba()()2000210Txyab===9 September 2002 Physics 217, Fall 2002 6Not everything is a gradient.Problem 1.30 in Griffiths: calculate1 212yx(2)(1)ba()() ()() ()2, from 1 1 0 to ˆˆ220, for 2 1 ,using both paths shown at right. Is the result independent of path?: 1 11, 2 10, so the result depends on path; this function isn't a gradient.dvy xyAnswer⋅===++∫bavl abxyPhysics 217, Fall 2002 9 September 2002(c) University of Rochester 39 September 2002 Physics 217, Fall 2002 7Divergence theorem The right-hand side is the fluxof the vector function vthrough the surface S. For example, if v were the velocity of a fluid, this would tell us the rate of flow of the fluid through the surface. The left hand side is the sum of all the sourcesof v within the volume. Again, in the example of fluid flow, this would be the sum of the output of the “faucets,” which in turn are places where the divergence of v is high.Proof – based on the nice one in Ch. 2 of Purcell – is provided on the chalkboard. ()VSddτ⋅=⋅∫∫vvav∇9 September 2002 Physics 217, Fall 2002 8Stokes’ theorem The right-hand side is the circulationof the vector function v on curve C. Again the name comes from the fluid-flow analogy. The left-hand side is the sum of the sourcesof circulation with the area bounded by C. If high divergence corresponds to a productive water faucet within V, high curl corresponds to a good stirring-rod within S. Again there’s a nice proof emphasizing this analogy, which I’ll show on the blackboard. Note: the orientation of the surface vector and the direction of the integration along the boundary should be consistent with the right-hand rule.()SCdd×=⋅∫∫va