Vector Calculus Theorems Green's, Stokes' and Divergence Frank Rooney, Sergio Loch and Jenny Dorrington In this worksheet we sum up the three big theorems of vector calculus. restart:with(plots):with(LinearAlgebra):

Green's theorem is just the two dimensional version of the fundamental theorem of calculus. The Fundamental Theorem is 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn whereas Greens theorem states thatLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn 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 The fundamental theorem converts integrals into functions and Greens theorem converts double integrals into line integrals. Consider the region R shown below with(plots): X := 5+4*cos(t): Y := 3+2*sin(t): p1 := plot([[X,Y,t=0..Pi], [X,Y,t=Pi..2*Pi]], linestyle=[1,2], color=[blue,red], scaling=constrained): p2 := plot({[[1,3],[1,0]],[[9,3],[9,0]]}, linestyle=2, color=blue): p21 := textplot({[1,-.5,`a`], [9,-.5,`b`], [5,5.5,`h`], [5,.6,`h`], [5.55,5.5,`(x)`], [5.55,.55,`(x)`]},font=[TIMES,ROMAN,10]): p22 := textplot({[5.25,5.28,`2`], [5.2,.35,`1`]}, font=[TIMES,ROMAN,8]): p23 := textplot([3,3,`R`], font=[TIMES,BOLD,14]): display({p1,p2,p21,p22},view=[0..10,-1..6], labels=[x,y], xtickmarks=[0], ytickmarks=[0], labelfont=[TIMES,ITALIC,12]);

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

And consider the integral 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 The last expression is derived using the one dimensional version of the fundamental theorem. Examining the last expression we see that the first term corresponds to the integral of F over the blue curve in the picture, moving from a to b, the second term is the the integral of F on the red curve. It is traditional however to traverse curves in the counterclockwise direction so the last integral comes just 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 Similarly if you consider the integral 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 referring to the picture below with(plots): X := 5+4*cos(t): Y := 3+2*sin(t): p3 := plot([[X,Y,t=-Pi/2..Pi/2], [X,Y,t=Pi/2..3*Pi/2]], linestyle=[1,2], color=[green,red], scaling=constrained): p4 := plot({[[5,5],[0,5]],[[5,1],[0,1]]}, linestyle=2, color=blue): p5 := textplot({[-.5,5,`c`], [-.5,1,`d`], [9.3,3,`k`], [.3,2,`k`], [9.9,3,`(y)`], [.9,2,`(y)`]},font=[TIMES,ROMAN,10]): p6 := textplot({[9.5,2.8,`2`], [.5,1.8,`1`]}, font=[TIMES,ROMAN,8]): p7 := textplot([3,3,`R`], font=[TIMES,BOLD,14]): display({p3,p4,p5,p6,p7},view=[-1..10,-.5..6], labels=[x,y], xtickmarks=[0], ytickmarks=[0],labelfont=[TIMES,ITALIC,12]);

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

Again the last integral is just the sum of the integral of G over the green curve moving from d to c followed by the integral of G over the red curve moving from c to d. So the integral becomes simply 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 Putting the two expressions together gives Green's Theorem 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

For the boundary of the square -1<x<1, -1<y<1, evaluate the line integral 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 First : Sketch the curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRitGK0YrRis= 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

On the curve C1 x=1, dx =0 and y varies from -1 to 1 On the curve C2 y=1, dy =0 and x varies from 1 to -1 On the curve C3 x=-1, dx =0 and y varies from 1 to -1 On the curve C4 y=-1, dy =0 and x varies from -1 to 1 So the line integral can be written as the sum of four integrals 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 which are easily evaluated as 20 On the other hand if we use Green's Theorem we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2MFErJkludGVncmFsO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSVmb3JtR1EncHJlZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGUi8lKG1pbnNpemVHRi4vJShtYXhzaXplR0YuLUYjNiQtRjk2MFEqJnVtaW51czA7RidGPC9GP1EmZmFsc2VGJy9GQkZpbi9GREZpbi9GRkZpbi9GSEZpbi9GSkZpbi9GTEZpbi9GTlEmaW5maXhGJy9GUVEwbWVkaXVtbWF0aHNwYWNlRicvRlRGY28vRlZRIjFGJy9GWFEpaW5maW5pdHlGJy1JI21uR0YkNiRGZm9GPC1GIzYjRmlvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZgcC1GOTYwUSJ+RidGPEZobkZqbkZbb0Zcb0Zdb0Zeb0Zfby9GTkYuRlBGU0Zlb0Znby1GNjYnLUY5NjBGO0Y8RmhuRmpuRkNGXG9GR0Zeb0Zfb0ZNRlBGU0Zlb0Znb0ZZRlxwRl5wRmFwRistSShtZmVuY2VkR0YkNiQtRiM2Jy1Gam82JFEiMkYnRjwtRiw2JVEieEYnRi9GMi1JJW1zdXBHRiQ2JS1GLDYlUSJ5RidGL0YyLUYjNiNGYHFGXnAtRjk2MFEiK0YnRjxGaG5Gam5GW29GXG9GXW9GXm9GX29GYG9GYm9GZG9GZW9GZ28tRmpvNiRRIjVGJ0Y8RjxGY3AtRjk2MFEwJkRpZmZlcmVudGlhbEQ7RidGPEY+RkEvRkRGQEZFL0ZIRkBGSUZLRmZwL0ZRRi4vRlRGLkZVRldGY3FGZHJGaXFGY3AtRjk2MFEiPUYnRjxGaG5Gam5GW29GXG9GXW9GXm9GX29GYG8vRlFRL3RoaWNrbWF0aHNwYWNlRicvRlRGX3NGZW9GZ29GY3BGNS1Gam82JFEjMTBGJ0Y8RmNwRmRyRmlxRltzLUZqbzYkUSMyMEYnRjw=

For the cardioid r = 1+cos\316\270 evaluate the line integral 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 First sketch the contour QyYtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIiLUklcGxvdEdGJTYnLCYiIiJGMC1JJGNvc0dGJTYjSSJ0R0YoRjBGNDssJEkjUGlHRiZGK0Y3L0knY29vcmRzR0YoSSZwb2xhckdGKC9JKHNjYWxpbmdHRihJLGNvbnN0cmFpbmVkR0YoRjA=

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Zio2I0kmdGhldGFHNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmKiosJiIiIkYsLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM5JEYsIiIjRi1GLC1JJHNpbkdGL0YyRiwsJiomRjVGLEYtRiwhIiIqJkYrRixGNUYsRjlGLEYsKiYsJiomRitGNEY1RjRGNComRitGLEYtRixGOUYsLCYqJEY1RjRGOUY+RixGLEYsRiVGJUYl

LCRJI1BpRyUqcHJvdGVjdGVkRyMhIzYiIiU=

Alternatively use Green's theorem, the line integral becomes 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 QyQtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1GJDYkLCYqJkkickdGKCIiIy1JJGNvc0dGJTYjSSZ0aGV0YUdGKCIiIiEiIkYuRjUvRi47IiIhLCZGNEY0RjBGNC9GMzssJEkjUGlHRiZGNUY9RjQ=

LCRJI1BpRyUqcHJvdGVjdGVkRyMhIzYiIiU=

(1) Evaluate the line integral in the first example over the boundary of the triangle formed by the lines x=1, y=1, x+y=1 (2) Using the the first quadrant section of the rose r = sin(3\316\270) evaluate 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

The Divergence Theorem allows us to calculate a surface integral by setting up and computing a triple integral. Equivalently, we can evaluate a triple integral by setting up and evaluating a surface integral. The Divergence Theorem states that the surface integral (flux) of a vector field F over a closed oriented surface S is equal to the triple integral of the divergence of the vector field over the closed region Q bounded by S, if the component functions of F have continuous partial derivatives. That is, \342\210\253S \342\210\253 F \302\267 N dS = \342\210\253 Q \342\210\253 \342\210\253 div F dV. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn The following example verifies the Divergence Theorem.

First define a vector field F and a surface S. field := <x,y,z>; surface := <x,y,4-x^2-y^2>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpV0xpPg==

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpbyJwJz4=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn We can visualize the interaction of the vector field and the surface by plotting them together: vfieldplot := fieldplot3d(field,x=-4..4,y=-4..4,z=-5..5, grid=[5, 5, 5]): surfaceparam := plot3d(surface,x=-2..2, y=-2..2): display({vfieldplot, surfaceparam}, axes=boxed);

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