CALCULUS IV MATH 241
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This 4 page Class Notes was uploaded by Claudine Friesen on Monday September 28, 2015. The Class Notes belongs to MATH 241 at University of Pennsylvania taught by N. Rimmer in Fall. Since its upload, it has received 9 views. For similar materials see /class/215395/math-241-university-of-pennsylvania in Mathematics (M) at University of Pennsylvania.
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Date Created: 09/28/15
Section 181 Contour Integrals F3 WWW Wm Let s go back to Section Parametric Curve xffvygf A B B C B 2 A B A A a Smooth b Piecewise id Simplc c Closed curve smooth cune closed bul nol curve simple Orientation of the curve m h zt lz C r r llm 3u qu Malhz lrkimmer SQ 131 Contour Integrals C parametric curve in the complex plane defined by x xt y y t a S t S b t real Now let zt xt iyta S t S b define C If C is piecewise srnooth we will now call it a An integral off z on C is called a The general If C is simple and To reinforce the positive symbol is closed we use orientation we use To evaluate the countour integral Intheformulaforfz IfZdZ replace z by C then dz for continuous f and smooth C m Math 241 Rimmer Evaluate 131 Contour Integrals Izzdz where C is de ned by z t 2 3t 2it 2 St S 2 C fzz2 SO fzt 1 3t2it3 Z1 dzz tdt dz Izzdz j f ztz tdt FE Math 241 Rimmer Evaluate 131 Contour Integrals x2 2 sz Where C is left half of the ellipse 7 L C 36 4 1 fromz 2 2i to z 2 2i 2 2 We need to parametrize C L L 1 36 4 Orientation S I S mt Math 241 Rimmer 131mm Odds and ends from Section 181 WW8 Complex functions can be thought of as Consider f z as a at the point z fz in yivxa y representation for the path of a particle in the flow gt rt 2 has vector T r t x t iy t that must coincide With the function fxtiyt 960 ytivxta yt dx dy 3 t and t 3 and 2 x y dt dt the solutions xt and y I are called the of the ow mg Math241 Rimmer Flnd the streamlines of the flow assomated W1thf z lZ 181 Cum mm f Z fz u u andv fiiiiT 3 3 l 7 d dt x A z z Q 3 Sep of Var Q dx r A the streamlines are centered at the origin v v v v mg 131 C m me Ll Continuing to View the complex function as a flow quot 39 V s we can now calculate and C positively oriented simple closed curve The around C measures the tendency of the ow to rotate C The 7 7 across C is the difference between the rate at which uid enters and the rate at which fluid leaves the region bounded by C PEI Math 2A1 7 Kim a y 181 Conmnr ln39eyals Boundin Theorem f iii on a if curve C 2 and L length of C for all z on C
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