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Study Guide for Calc 3

by: Nikki Bee

Study Guide for Calc 3 MA 261

Marketplace > Purdue University > Mathematics (M) > MA 261 > Study Guide for Calc 3
Nikki Bee

GPA 4.0
Calculus 3

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Will help for your final exam for calc 3
Calculus 3
Study Guide
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This 6 page Study Guide was uploaded by Nikki Bee on Wednesday June 10, 2015. The Study Guide belongs to MA 261 at Purdue University taught by in Spring 2015. Since its upload, it has received 54 views. For similar materials see Calculus 3 in Mathematics (M) at Purdue University.

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Date Created: 06/10/15
MA 261 Fall 2012 Study Guide 3 You also need Study Guides 1 and 2 for the Final Exam Line integral of a function f 13 y 2 along C parameterized by 1 2 3375 y yt z 2t and a S t S b is C fw7y7z ds bfat ya 20 dt independent of orientation of C other properties and applications of line integrals of f Remarks a 0 fx y 2 ds is sometimes called the line integral of f with respect to arc length b emsmas bfltxlttgtylttgtzlttgtgtx39lttgtdt c Cfwyzdy bfltwlttgtylttgtzlttgtgty39lttgtdt d C mews bfltxlttgtylttgtzlttgtgtz39lttgtdt Line integral of vector eld F a y 2 along C parameterized by t and a S t S b is given by b F df Ff t m dt C 1 depends on orientation of C other properties and applications of line integrals of f Connection between line integral of vector elds and line integral of functions fFdFF Tds C C Where 39f is the unit tangent vector to the curve C If F013 3 Z PC7341 2 iUr 62013731 Zi R613 3 Z 12 then f df PwyzdwQwyzdyRwyzdz C C Work FquotdI quot C FUNDAMENTAL THEOREM OF CALCULUS FOR LINE INTEGRALS Vfdf f Fb f Fa c C rb 8Q8P 6 A vector eld 2 135 y i Qayjis conservative ie F Vf if E 8y how to determine a potential function f if Vf 8Q 8P 7 GREEN S THEOREM Pxy d3 Qxy dy 8 8 dA C boundary of D C D 33 y As a consequence of Green s Theorem one has 1 xdy ydxzf xdyz f ydxAreaD 2 C C C a 7 8 7 8 8 Del Operator 1 a yJ a f E 8 8 8 lFZV F cur x 833 ay 82 P Q R Properties of curl and divergence O 12 if FOE y Z PC7341 2 iUr 62013731 2 R613 3 127 then 8P 8Q 8R and leF VFa xa yg i If curl F 6 then F is a conservative vector eld ie7 Vf ii If curl F 6 then F is irrotational if div F 07 then F is incompressible 9 Parametric surface S Hum ltxuvyuvzuv Where u39v E D V D L L AK X Normal vector to surface S 2 Eu gtlt ECU tangent planes and normal lines to parametric surfaces 10 Surface area of a surface S 11 12 i AS Iquotu gtlt FvdA D ii If S is the graph of z ha y above D then AS 1 Sh3 332 8ho y2 dA D Remark dS lfu gtlt EU dA differential of surface area While dS 17 X EU dA The surface integral of f 13 y 2 over the surface S i fayzgtd8 flt uvgtlr ugtltmdA ii If S is the graph of z ha y above D then dS h aha 2 ME 2dA Ami2 Dfltxy xygtgt 1lt M lt ygt The surface integral of F over the surface S recall dS 171 X EU dA fSFd BFmmm fSF d AF dszf uxmm If S is the graph of z ha y above D With oriented upward and F P Q R then d D P Qg ZR M i Connection between surface integral of a vector eld and a function fgFdsS ds The above gives another way to compute S F ZS ii F dS dS 1 of F across the surface S s s n S 13 STOKES THEOREM F if curl d recall7 curlf V X C S n F dfquot circulation Of F around C C 14 THE DIVERGENCE THEOREMGAUSS THEOREM F d divf dV 5 E recall7 divf V n 34 15 Summary of Line Integrals and Surface Integrals LINE INTEGRALS SURFACE INTEGRALS CFtwherea t b S Fu39v Where u39v E D d8 l dt differential of arc length dS lfu gtlt EU dA differential of surface area d8 length of C 0 dS 2 surface area of S S C m y 2 ds mt lf tl dt independent of orientation of C fS 357972 ff u7v If x m dA independent of normal vector R r m dt ds a x a M fc df fab m m dt depends on orientation of C fS 39dng uw E u gtlt EU dA depends on normal vector LFdFLFTds The circulation of F around C fS dsSF dS The flux of F across S in direction 16 Integration Theorems b FUNDAMENTAL THEOREM OF CALCULUS F 3 d3 Fb Fa b FUNDAMENTAL THEOREM OF CALCULUS FOR LINE INTEGRALS V f df f E b f E a C a rb GREEN S THEOREM D STOKES THEOREM curlF d F df S C DIVERGENCE THEOREM divf dV 2 f F d E s n S 34


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