MAT2500-01/02 20S Test 4 Print Name (Last, First) _________________________________|__Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use EQUAL SIGNS and arrows when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation IF appropriate). Indicate where technology is used and what type (Maple, GC). Only use technology to CHECK hand calculations, not substitute for them. Explain in as many words as possible everything you are doing!
1. Given 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 over the closed curve which is the counterclockwise oriented semicircle 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 completed by the diameter line segment along the x axis.a) Evaluate the line integral directly using a polar coordinate parametrization of the semicircle. Be sure to state the simplified integrand before using technology to evaluate the iterated integral.b) Use Green's Theorem to evaluate the equivalent double integral, again in polar coordinates. Be sure to state the simplified integrand before using technology to evaluate the iterated integral.
2. Given the two parametrized curves 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 and 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 , and the vector field 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.a) Set up and simplify the line integrals of the vector field over each of these two paths between the same endpoints, then use technology to evaluate them.b) Verify the differential condition that this vector field admit a scalar potential.c) Solve the equations needed to construct a scalar potential.d) Use the scalar potential to evaluate the line integral between these endpoints.3. Consider the radial vector field 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 in the plane, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 is a constant. Show all the steps in evaluating and simplifying the partial derivatives.a) Evaluate the magnitude of this vector field.b) Evaluate and simplify 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.c) Evaluate and simplify div(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) .d) Solve the equations which determine a potential function for this vector field in the plane.e) Notice that this vector field is not defined at the origin where its magnitude has an infinite limit. Evaluate the (scalar) line integral of its outward normal component which appears in the Gauss version of Green's theorem 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
for the region R inside a counterclockwise oriented circle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIrRidGPi8lJmZlbmNlR0ZCLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlotRiw2JS1GLzYlUSJ5RidGMkY1RjhGQy1GRzYtUSI9RidGPkZKRkxGTkZQRlJGVEZWL0ZZUSwwLjI3Nzc3NzhlbUYnL0ZmbkZgby1GLDYlLUYvNiVRImFGJ0YyRjVGOEZDLUYvNiNRIUYnRkBGPg== to see the consequence of this single point for the field. How does this compare to the double integral side of the Gauss-Green theorem?pledgeWhen you have completed the exam, please read and sign the dr bob integrity pledge and and scan this test sheet as a cover first page in the PDF scan of your lined paper hand work all on separate sheets."During this examination, all work has been my own. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."Signature: Date: solution1. Green's TheoremThis closed loop has two piecewise contributions, first the line segment along the horizontal axis, then the upper semicircle returning to the origin.We have done this so many times, the upper semicircle is 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 in polar coordinates.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) line integral top: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 integral bottom and total: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b) double integral: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JSFH2. 3d line integralboth curves start and end in the same points. They happen to belong to the same surface of revolution.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The two curves: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The scalar potential fieldLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGPS1GLDYlUSJ6RidGL0YyRkFGQS1GPjYtUSomY29sb25lcTtGJ0ZBRkMvRkdGRUZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGXW8tSSZtZnJhY0dGJDYoLUkjbW5HRiQ2JFEiMUYnRkEtRiM2JC1GY282JFEiMkYnRkFGQS8lLmxpbmV0aGlja25lc3NHRmVvLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl9wLyUpYmV2ZWxsZWRHRkUtRj42LVEnJnNkb3Q7RidGQUZDRltvRkhGSkZMRk5GUEZSL0ZWRlQtRjY2JC1GIzYsLUklbXN1cEdGJDYlLUY2NiQtRiM2JkY6LUY+Ni1RKCZtaW51cztGJ0ZBRkNGW29GSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRmdxRlhGQUZBRmZvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GPjYtUSIrRidGQUZDRltvRkhGSkZMRk5GUEZmcUZocUZob0ZkcC1GXXE2JS1GNjYkLUYjNiZGOkZcckZYRkFGQUZmb0ZpcUZcci1GY282JFEiM0YnRkEtRj42LVEifkYnRkFGQ0Zbb0ZIRkpGTEZORlBGUkZncC1GXXE2JUZlbkZmb0ZpcUZBRkEtRj42LVEiO0YnRkFGQ0ZGRkhGSkZMRk5GUEZSRl5vLUYsNiVRJ2V4cGFuZEYnRi9GMi1GNjYkLUYjNiZGKy1GNjYkLUYjNilGOkY9RlhGPUZlbi8lK2V4ZWN1dGFibGVHRkVGQUZBRlt0RkFGQUZbdEZBLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2JVEkZWZmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSZtZnJhY0dGJDYoLUYjNiQtSSNtbkdGJDYkUSIzRidGOUY5LUZVNiRRIjJGJ0Y5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zqbi8lKWJldmVsbGVkR0Y9LUY2Ni1RMSZJbnZpc2libGVUaW1lcztGJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GY28tSSVtc3VwR0YkNiUtRiw2JVEieEYnRi9GMkZYLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUScmcGx1cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GYnAtRiw2JVEieUYnRi9GMkZfb0Zob0ZecC1GUDYoRlRGWEZlbkZobkZbb0Zdb0Zfby1GZm82JUZkcEZYRltwRl5wRmdwRl9vLUZmbzYlLUYsNiVRInpGJ0YvRjJGWEZbcC1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIRmJvRk0tSShtZmVuY2VkR0YkNiYtRiM2LC1GUDYoLUY2Ni1RKyZQYXJ0aWFsRDtGJ0Y5L0Y8USZ1bnNldEYnL0Y/Rl9yL0ZBRl9yL0ZDRl9yL0ZFRl9yL0ZHRl9yL0ZJRl9yRmJvRmRvLUYjNiYtRjY2LUZdckY5RjtGPkZARkJGREZGRkhGYm9GZG8tRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZib0Zkb0Zob0Y5RmVuRmhuRltvRl1vLUZlcTYkLUYjNiVGK0YvRjJGOS1GNjYtUSIsRidGOUY7RmNxRkBGQkZERkZGSEZiby9GTlEsMC4zMzMzMzMzZW1GJy1GUDYoRltyLUYjNiZGaHJGanJGZHBGOUZlbkZobkZbb0Zdby1GZXE2JC1GIzYkRitGOUY5RmFzLUZQNihGW3ItRiM2JkZockZqckZdcUY5RmVuRmhuRltvRl1vRmpzLyUrZXhlY3V0YWJsZUdGPUY5RjkvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRmJ0Rjk=The gradient vector field: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first line integral: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second line integral: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potential difference: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. line charge in 3d: 2d cross sectionFor those interested in physical applications:
An inverse square force field like the electric or gravitational field generated by an infinite line of charge or mass has a cylindrical symmetric force field inversely proportional to the distance from the axis, i.e., 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 in cylindrical coordinates. Looking at a cross section of this field orthogonal to the symmetry access leads to this logarithmic potential function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEnJnByb3A7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUSNsbkYnLyUnaXRhbGljR0Y0Ri8tSShtZmVuY2VkR0YkNiQtRiM2JS1GRzYlUSJyRicvRktRJXRydWVGJy9GMFEnaXRhbGljRicvJStleGVjdXRhYmxlR0Y0Ri9GL0ZYRi8=.
Both its gradient and curl vanish, EXCEPT at the origin where the source of the field lies and the field strength is infinite. Therefore Green's theorem does not apply to any curves which loop around the origin. This exercise demonstrates this fact. For such curves, the integral is just a constant independent of the path since it is a measure of the charge or mass inside the curve.applicationSearch on: electric field of an infinite line of charge. For example:
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.htmlThe fact that the divergence integral over a circle about the origin should equal a nonzero result when the divergence is zero at all points except the origin is a motivation for the introduction of the Dirac delta function, which is zero except at one point but whose integral over a region containing the point is 1 (this too is a limiting concept like the fundamental ideas of calculus).Here LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5 is set, but all results are proportional to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== for the test.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0ZFRkFGQS1GPjYtUSomY29sb25lcTtGJ0ZBRkMvRkdGRUZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGXG8tSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVEia0YnRi9GMkYvRjItRiM2JS1JI21uR0YkNiRRIjJGJ0ZBRi9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYXAvJSliZXZlbGxlZEdGRS1GPjYtUSJ+RidGQUZDRmpuRkhGSkZMRk5GUEZSL0ZWRlQtRiw2JVEjbG5GJy9GMEZFRkEtRjY2JC1GIzYnLUklbXN1cEdGJDYlRjpGZm8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY+Ni1RIitGJ0ZBRkNGam5GSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRlxyLUZjcTYlRlhGZm9GZXFGZW5GQUZBLUY+Ni1RIjtGJ0ZBRkNGRkZIRkpGTEZORlBGUkZdby1GLDYlUSZncmFkZkYnRi9GMkY1RmduLUY2NiYtRiM2KS1GX282KC1GPjYtUSsmUGFydGlhbEQ7RidGQS9GRFEmdW5zZXRGJy9GR0Zgcy9GSUZgcy9GS0Zgcy9GTUZgcy9GT0Zgcy9GUUZgc0ZSRmlwLUYjNiYtRj42LUZec0ZBRkNGam5GSEZKRkxGTkZQRlJGaXBGZnBGOkZBRlxwRl9wRmJwRmRwLUY2NiQtRiM2JkYrLUY2NiQtRiM2J0Y6Rj1GWEYvRjJGQUYvRjJGQUY9LUZfbzYoRlxzLUYjNiZGaXNGZnBGWEZBRlxwRl9wRmJwRmRwLUY2NiQtRiM2JUYrLUY2NiQtRiM2JkY6Rj1GWEZBRkFGQUZBRmVuRkFGQS8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RictRj42LVEiOkYnRkFGQ0ZqbkZIRkpGTEZORlBGW29GXW9GZnBGY3JGNUZlbkZBvanishing curl: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integrating to get the potential.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JVEkcG90RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRjY2LVErJkludGVncmFsO0YnRjlGO0Y+L0ZBRjFGQi9GRUYxRkZGSC9GS1EmMC4wZW1GJy9GTkZVLUkmbWZyYWNHRiQ2KC1GIzYqLUYsNiVRImtGJ0YvRjItRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSEZURlYtRiw2JVEieEYnRi9GMi8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHRj0vJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjktRiM2Ki1JJW1zdXBHRiQ2JUZcby1JI21uR0YkNiRRIjJGJ0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUScmcGx1cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GaXAtRlxwNiUtRiw2JVEieUYnRi9GMkZecEZicEZfb0Zib0Zkb0Zmb0Y5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZlcS8lKWJldmVsbGVkR0Y9LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGVEZWLUY2Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjkvRjxRJnVuc2V0RicvRj9GYXIvRkFGYXIvRkNGYXIvRkVGYXIvRkdGYXIvRklGYXJGVEZWRlxvLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGaHBGanAtRiw2JVEiQ0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlRl1xRmJvRjlGOS1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIRlRGTS1GWDYoLUY2Ni1RKyZQYXJ0aWFsRDtGJ0Y5RmByRmJyRmNyRmRyRmVyRmZyRmdyRlRGVi1GIzYmLUY2Ni1GW3RGOUY7Rj5GQEZCRkRGRkZIRlRGVkZqcUZdcUY5RmBxRmNxRmZxRmhxLUZfczYkLUYjNigtRiw2JVEiJUYnRi9GMkYvL0Zgb1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRjEvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjFGMkY5LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUZYNigtRiM2KkZmbkZpbkZdcUZfb0Zib0Zkb0Zmb0Y5RmlvRmBxRmNxRmZxRmhxRmNzRltzLUY2Ni1RIidGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRidGVkZec0ZddS1GX3A2JEZkcEY5RmNzRltzRl5zRl11RltzRmJvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYwLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0ZFRkFGQS1GPjYtUSomY29sb25lcTtGJ0ZBRkMvRkdGRUZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGXG8tSSZtZnJhY0dGJDYoLUYjNiYtRiw2JVEia0YnRi9GMi1GPjYtUSJ+RidGQUZDRmpuRkhGSkZMRk5GUEZSL0ZWRlQtRjY2JkY4RkEvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRkEtRiM2Jy1JJW1zdXBHRiQ2JUY6LUYjNiUtSSNtbkdGJDYkUSIyRidGQUYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY+Ni1RIitGJ0ZBRkNGam5GSEZKRkxGTkZQL0ZTUSwwLjIyMjIyMjJlbUYnL0ZWRmRxLUZlcDYlRlhGZ3BGXXFGL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zdci8lKWJldmVsbGVkR0ZFLUY+Ni1RIjpGJ0ZBRkNGam5GSEZKRkxGTkZQRltvRl1vRmZvRitGNS1GPjYtUSI7RidGQUZDRkZGSEZKRkxGTkZQRlJGXW8tRiw2JVElRm1hZ0YnRi9GMkZnbi1GX282KC1GIzYlRmNvRi9GMi1GIzYnLUYsNiNRIUYnLUYjNiVGYXMtSSZtc3FydEdGJDYjLUYjNiZGZHBGYHFGZnFGQUZBRmFzRi9GMkZocUZbckZeckZgckZlbkZBdivergence vanishes.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e) Notice that this vector field is not defined at the origin where it has an infinite limit. Evaluate its line integral around a counterclockwise oriented circle 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 to see the consequence of this single point for the field.The outward unit normal is along the radial direction proportional to the position vector, derived again from the unit tangent and a 90 degree rotation.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 normal in radial direction: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=