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# Multivariable Calculus Engrs MATH 1920

Cornell

GPA 3.75

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This 25 page Class Notes was uploaded by Sandrine Wehner II on Saturday September 26, 2015. The Class Notes belongs to MATH 1920 at Cornell University taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/214326/math-1920-cornell-university in Mathematics (M) at Cornell University.

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Date Created: 09/26/15

1 a 6 points For What values of c is the eld F y oz il a j k conservative Mark the correct statement below You do not need to justify your answer for every value of c Ell JED for no value of c y b 13 points The eld F zcosa i ley 1nz j 2 sina k is conservative Find a potential function for F 5 a 6 points Consider the equation 45 3 in spherical coordinates Mark the correct statement below You do not need to justify your answer II The equation qb 17 describes a Sphere III The equation go g describes a cylinder CI The equation q 17r describes a cone D The equation Q5 2 describes a plane b 12 points A circular cylindrical hole is bored through the solid sphere of radius 2 and center 0 0 0 The wall of the cylindrical hole is on the cylinder 02 y2 a2 Compute the volume of the remaining solid in terms of 1 Find the radius a of the cylinder such that the removed volume is 7 times the remaining volume 18 points Match each shaded region With the integral that computes the area of the region Every region matches exactly one double integral The regions and the integrals are given below You do not need to justify your answers 2 arccos The integral f r d dr matches Figure 1 o 27r 0 The integral f 7quot dram matches Figure 0 go IL 1 The integral 3 7 7quot drd6 matches Figure 0 1 2 0059 1 arcsin39r The integral r d dr matches Figure 0 0 321 1cos6 The integral f 7 drd6 matches Figure 0 cos6 arccos39r39 The integral 7quot d dr matches Figure 0 arcsin739 Figure B Figure C 2 1 Figure Figure E 03 Consider the vector eld 9 My 317 where r 2 y2 a 5 pts Calculate the divergence of F7 div F b 10 pts For a closed curve C that does not go around the origin7 evaluate the outward ux 550 F nds Can Green7s Theorem be used here Explain your answer c 10 pts For the unit circle 0 around the origin7 nd the outward ux f0 F nds Can Green7s Theorem be used here Explain your answer b 12 points Use Lagrange Multipliers in order to nd the absolute maximum value of the function 1 y z xyz2 on the sphere 12 y2 22 i 4 5 20 puiursb Find the lJl39IClJJJVlDJ x 11mm of a lum llon rmjnod by right p439rims I39ty mm llu vllipmhl w 15 I 1 win Ur k llsih39 mi 110m xiv pztrnhtuhi y iii in plane Thr wirv mm mm 71 3 u 2 n 1 an integral in llh mm of mi in mi umd nnl mth m imgm in UN i i and inside Lin rink plimn umsidri he mgiun 1 mm is mmth Lui Link iii M SimLi Hm rirclr um fhi39 l lffli v 15m Find the HhL nK FUun plums h mm a iiuumi inwng ii i mm or miblu imvgn s m um mu iii n wmi mi nnm Ur il gm nn m m Yriru H mm uncng or ii mini I dm vlc iniogmls lur um mm m R wiui uii imm uf ilich mLiuu d a ulcuhm m imii ui E C39unsidui uic Uiplc inu gml ffV p quaiwzpz iii112 u i a Clinngi Lhr mm M imcgmiimi m 1217M m l Ewimm Hm iuiwgmi Calculate l hu LirEHl Htm of the cl F iiii mmu Ulnclmise39 mound uic hounth uf mi uiiiiigiewmi whim o u 1 1 and 20 We aaxumi39 rim 3 C uusulcr u pmlmiuid but mm 1mm 2 1 and Mir 2 l the Inn in unL insidr he cviiumx Lir n m llL a Cumqu H r Valium 01 Ii i L Sci b Lr Find b it HIL wiumu oi R is A quzil w 31 UN whim ni Llw hm Dot Product 123 continued and The Cross Product 124 Dot Product 123 continued and The Cross Product 124 Last time in 192 gt We learned about the dot product of two vectors and its application to work and flux Recall Dot Product 123 continued and The Cross Product 124 Last time in 192 gt We learned about the dot product of two vectors and its application to work and flux Recall gt Theorem Let 9 be the angle between two vectors L7 and 7 then a u 7cos0l ll7l Dot Product 123 continued and The Cross Product 124 Last time in 1920 gt We learned about the dot product of two vectors and its application to work and flux Recall gt Theorem Let 9 be the angle between two vectors L7 and 7 then u i7cos0lL7lli7l gt Let39s revisit the water flowing down the pipe Dot Product 123 continued and The Cross Product 124 Is the amount of water that crosses AP equal to the what crosses As in one sec Is Apnj 7 Agn 7 Let39s look at the geometry board sketches Dot Product 123 continued and The Cross Product 124 Writing L7 as a sum of components Use a sketch gt Problem Write L7lt 147 6 gt as the sum of a vector parallel to 7lt 269 gt and a vector perpendicular to 7 Dot Product 123 continued and The Cross Product 124 Writing L7 as a sum of components Use a sketch gt Problem Write L7lt 147 6 gt as the sum of a vector parallel to 7lt 269 gt and a vector perpendicular to 7 gt The scalar component of L7 in the direction of i7 is 3 mo m1246 6x9 80 x53x121 121 Dot Product 123 continued and The Cross Product 124 Writing L7 as a sum of components Use a sketch gt Problem Write L7lt 147 6 gt as the sum of a vector parallel to 7lt 269 gt and a vector perpendicular to 7 gt The scalar component of L7 in the direction of i7 is 3 mo m1246 6x9 80 x53x121 121 gt The vector projection of L7 onto 7 is ro39 Eil lcos0j7780 lt 27679gt p 1quot lvl 121 121 39 Dot Product 123 continued and The Cross Product 124 Writing L7 as a sum of components Use a sketch gt Problem Write L7lt 147 6 gt as the sum of a vector parallel to 7lt 269 gt and a vector perpendicular to 7 gt The scalar component of L7 in the direction of i7 is 3 mo m1246 6x9 80 V53V121 121 gt The vector projection of L7 onto 7 is ro39 L77 l lcos0j 7 780 lt 27679 gt p 1quot T 7 T V121 V121 39 gt The vectors we want are projVU and L77 projg See page 860 for the computation that shows that their dot product is 0 So a 80 lt27679gt 8O lt27679gt u7 lt146gt77 121 V121 V121 V121 Dot Product 123 continued and The Cross Product 124 The Cross ProductWhat you need to Know gt How to compute it Use the determinant trick Example L7 3i1j and i7j2k Find UXV Dot Product 123 continued and The Cross Product 124 The Cross ProductWhat you need to Know gt How to compute it Use the determinant trick Example L7 3i1j and i7j2k Find UXV 277 67 31 Dot Product 123 continued and The Cross Product 124 The Cross ProductWhat you need to Know gt How to compute it Use the determinant trick Example L7 371jand i739 21 Find UXV gt 277 67 3 gt How to visualize it Sketch The vector points in the direction determined but the right hand rule curling the fingers from L7 to i7 your thumb points in the direction of UXV Dot Product 123 continued and The Cross Product 124 The Cross ProductWhat you need to Know gt How to compute it yse the determinant trick Example L7 3i1j and i7j2k Find UXV gt 277 67 3 gt How to visualize it Sketch The vector points in the direction determined but the right hand rule curling the fingers from L7 to i7 your thumb points in the direction of UXV gt What it is used for Dot Product 123 continued and The Cross Product 124 The Cross ProductWhat you need to Know gt How to compute it Else the determinant trick Example L7 3i1j and i7j2k Find UXV gt 277 67 3 gt How to visualize it Sketch The vector points in the direction determined but the right hand rule curling the fingers from L7 to i7 your thumb points in the direction of UXV gt What it is used for gt L7Xi7 is a vector that is perpendicular to the plane determined by L7 and i7 Dot Product 123 continued and The Cross Product 124 The Cross ProductWhat you need to Know gt How to compute it Else the determinant trick Example L7 3i1j and i7j2k Find UXV 277 67 31 How to visualize it Sketch The vector points in the direction determined but the right hand rule curling the fingers from L7 to i7 your thumb points in the direction of UXV What it is used for V V V V L7Xi7 is a vector that is perpendicular to the plane determined by L7 and i7 V lts length is the area of the parallelogram spanned by L and i7 Dot Product 123 continued and The Cross Product 124 Flux across a parallelogram an important example gt Fluid is flowing with constant velocity i7 across the parallelogram spanned by 3 and b Sketch Dot Product 123 continued and The Cross Product 124 Flux across a parallelogram an important example gt Fluid is flowing with constant velocity i7 across the parallelogram spanned by 3 and b Sketch gt The rate at which the fluid flows across the parallelogram is Dot Product 123 continued and The Cross Product 124 Flux across a parallelogram an important example gt Fluid is flowing with constant velocity i7 across the parallelogram spanned by 3 and b Sketch gt The rate at which the fluid flows across the parallelogram is gt the product of the area of the parallelogram lEXBl and the normal component of i7 7 33 So the flux is a Dot Product 123 continued and The Cross Product 124 Flux across a parallelogram an important example gt Fluid is flowing with constant velocity i7 across the parallelogram spanned by 3 and b Sketch gt The rate at which the fluid flows across the parallelogram is gt the product of the area of the parallelogram lEXBl and the normal component of i7 7 35 So the flux is a U l mi X 7 isti maxi 3 E X gt The flux is the triple scalar product of i7 3 and Dot Product 123 continued and The Cross Product 124

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