Vector Calculus TheoremsGreen's, Stokes' and DivergenceFrank Rooney, Sergio Loch and Jenny DorringtonIn this worksheet we sum up the three big theorems of vector calculus. restart:with(plots):with(LinearAlgebra):Green's TheoremProof of Green's TheoremGreen's theorem is just the two dimensional version of the fundamental theorem of calculus. The Fundamental Theorem 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Ynwhereas Greens theorem states 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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 belowwith(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]);And consider the 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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 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Similarly if you consider the 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 referring to the picture belowwith(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]);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 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 the two expressions together gives Green's 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnA Simple Example
For the boundary of the square -1<x<1, -1<y<1, evaluate the line integralLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2MFErJkludGVncmFsO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSVmb3JtR1EncHJlZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGUi8lKG1pbnNpemVHRi4vJShtYXhzaXplR0YuLUYjNiMtRiw2JVEiQ0YnRi9GMkYrLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZqbkYrLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtbkdGJDYkUSIzRidGPC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiMtRmNvNiRRIjJGJ0Y8RmhuLUY5NjBRIn5GJ0Y8L0Y/USZmYWxzZUYnL0ZCRjEvRkRGZXAvRkZGZXAvRkhGZXAvRkpGZXAvRkxGZXAvRk5RJmluZml4RidGUC9GVFEzdmVyeXRoaWNrbWF0aHNwYWNlRicvRlZRIjFGJy9GWFEpaW5maW5pdHlGJy1GOTYwUSomdW1pbnVzMDtGJ0Y8RmRwL0ZCRmVwRmdwRmhwRmlwRmpwRltxRlxxL0ZRUTBtZWRpdW1tYXRoc3BhY2VGJy9GVEZpcUZgcUZicS1GY282JFEiNUYnRjwtRiw2JVEieUYnRi9GMkY8LUY5NjBRMCZEaWZmZXJlbnRpYWxEO0YnRjxGPkZBL0ZERkBGRS9GSEZARklGSy9GTkYuL0ZRRi4vRlRGLkZVRldGaW8tRjk2MFEiK0YnRjxGZHBGZ3FGZ3BGaHBGaXBGanBGW3FGXHFGaHFGanFGYHFGYnFGZm9GKy1GZ282JUZeckZccEZobi1GLDYlUSNkeUYnRi9GMg==First : Sketch the curveLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRitGK0YrRis=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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
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A More Complicated ExampleFor the cardioid r = 1+cos\316\270 evaluate the line integral
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 sketch the contourQyYtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJnBsb3RzR0YlISIiLUklcGxvdEdGJTYnLCYiIiJGMC1JJGNvc0dGJTYjSSJ0R0YoRjBGNDssJEkjUGlHRiZGK0Y3L0knY29vcmRzR0YoSSZwb2xhckdGKC9JKHNjYWxpbmdHRihJLGNvbnN0cmFpbmVkR0YoRjA=To evaluate this we need to convert the cartesian coordinates to polars and use the equation for the cardioid and then integrateQzI+SSd4ZnVuY3RHNiJmKjYjSSZ0aGV0YUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYsJiIiIkYuLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM5JEYuRi5GL0YuRiVGJUYlRi4+SSd5ZnVuY3RHRiVmKkYnRiVGKUYlKiZGLUYuLUkkc2luR0YxRjRGLkYlRiVGJUYuPkkjZHhHRiUtSSJER0YlNiNGJEYuPkkjZHlHRiUtRj82I0Y3Ri4+SSppbnRlZ3JhbmRHRiUsJiooRiRGLkY3Ri5GPUYuRi4qJiwmKiRGNyIiI0ZMRiQhIiJGLkZCRi5GLkZNPkklZmludEdGJWYqRidGJUYpRiUsJioqRi1GTEYvRi5GOkYuLCYqJkY6Ri5GL0YuRk1GOUZNRi5GLiomLCYqJkYtRkxGOkZMRkxGLEZNRi4sJiokRjpGTEZNRixGLkYuRi5GJUYlRiVGLi9JJmludGVnR0YlLUkodW5hcHBseUdGJTYkLUZGRidGKEZNLUkkaW50R0YlNiQtRk9GJy9GKDssJEkjUGlHRjJGTUZhb0YuAlternatively use Green's theorem, the line integral becomes
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QyQtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1GJDYkLCYqJkkickdGKCIiIy1JJGNvc0dGJTYjSSZ0aGV0YUdGKCIiIiEiIkYuRjUvRi47IiIhLCZGNEY0RjBGNC9GMzssJEkjUGlHRiZGNUY9RjQ=Exercises(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) 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 Divergence TheoremThe 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnThe following example verifies the Divergence Theorem.ExampleFirst define a vector field F and a surface S.field := <x,y,z>;
surface := <x,y,4-x^2-y^2>;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);