Multivariable Calculus MATH 222
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This 4 page Study Guide was uploaded by Lauriane Brown on Wednesday October 28, 2015. The Study Guide belongs to MATH 222 at Vassar College taught by Staff in Fall. Since its upload, it has received 36 views. For similar materials see /class/230533/math-222-vassar-college in Mathematics (M) at Vassar College.
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Date Created: 10/28/15
Test 2 Review SheetiMath 222 Prof Frank Spring 2005 This exam will cover sections 41 44 51 55 61 62 71 72 13 total General principles to guide your study 1 Review your notes for each section of the book and the comments in the study guide for that section Make a note of every important fact de nition and theorem from that section that you feel you should memorize 2 Go over your HW assignments and make sure that you understand all of the problems as well as the related problems in the text Pay close attention to the ones you missed the rst time around 3 Think about the questions listed below after you have completed your review Try and gure them out without referring to your notes Warning some of the following questions are much more vague and open ended than what will be on your exam Consider them food for thought 1 What is the difference between real valued functions and vector valued functions What are the two main types of vector valued functions we ve studied and how can you graph them 2 Suppose that c is a path for which c O ls there a force acting on c 3 Suppose c is a unit speed path What does this mean De ne the vectors in the Frenet Frame of c 4 Suppose c is a unit speed path ls it possible for Bt 7 0 but Nt 0 Explain twice once with mathematical expressions and again on an intuitive level 5 Prove that the vectors in the Frenet Frame must be othogonal to their derivatives 6 De ne the curvature function kt and the torsion function 7t Explain what they tell you about your unit speed curve 7 If Ct is a unit speed path with zero curvature what does this imply about c If instead it has zero torsion what does that imply about U 8 What does it mean to say that F is a gradient vector eld ls every vector eld a gradient vector eld Either prove or nd an example to the contrary 9 Suppose F R3 a R3 is a vector eld and that c is a ow line for F Give the domain and range of c and the precise equation connecting it to F Explain the equation in a sentence 10 True or false Give a proof if true and a counterexample if false a If the divergence of a vector eld is zero then the curl must be nonzero b If the divergence of a vector eld is zero then the curl must be zero also c If F is a C1 gradient vector eld its divergence is zero d If F is a C1 gradient vector eld its curl is zero 11 Compare and contrast the methods of obtaining volumes via Riemann sums and Cavalieri s principle Draw pictures to illustrate your discussion 12 Explain the two things in Fubini s theorem that make computing double and triple integrals easier 82 13 Let D w x cd Showthat m fac7 mm fbd 7mm 14 Let D be the region given by 2 y2 lt 4 Without doing any calculations explain why mdA 0 D 15 Let D be a region in R2 and let W be a solid in R3 What do the integrals 1A and D dV represent 16 Explain the mean value inequality from page 3527 using sketches to prove your point 17 What does it mean for a function to be one to one For what theorem that we are currently studying is this property important 18 Write down the equations for converting polar coordinates into rectangular coordinates 8 Show how to compute the Jacobian determinant 82x73 needed for the change of variables T7 formula 19 Write down the equations for converting cylindrical coordinates into rectangular coordi 7 972 W needed for the change of nates Show how to compute the Jacobian determinant variables formula 20 Write down the equations for converting spherical coordinates into rectangular coordinates 137 297 2 897 07 gtgt formula 21 Say a few words about why the Jacobian determinant appears in the change of variables formula What would happen if it wasn t there 22 Suppose c 11 a R3 is a path that we imagine is made of wire7 and suppose that f R3 a R represents the density of the wire What is the interpretation of the path Show how to compute the Jacobian determinant needed for the change of variables integralfds C 23 Suppose that c 071 a R3 is a path that a bumblebee follows Suppose F R3 a R3 is a force eld7 such as gravity7 that acts on the bumblebee Note that the force eld doesn t necessarily cause the bee to follow the path c7 but it does have an effect on it What is the interpretation of the line integral F ds 24 Let c be a path in the plane R2 and let 1 R2 a R ls there an area interpretation of 1 ds Sketch something to illuminate your answer 25 Lcet Tt be the unit tangent vector to the path Ct at time t Prove that F ds F T d37 where the integral on the right is a path integral 26 Porove that the line integral of a gradient vector eld depends only on its endpoints 27 Prove that the integral of a gradient vector eld over a closed curve is O 28 Let c 11 a R3 denote a path The path p 719701 a R3 given by pt 17 de nes the same curve but traversed in the opposite direction Convice yourself that Fds7Fdsandfdsfds p c p c AA Test 1 Review SheetiMath 222 Prof Frank Spring 2005 This exam will cover sections 11 through 14 21 through 26 and 31 through 34 fourteen total General principles to guide your study 1 Review your notes for each section of the book and the comments in the study guide for that section Make a note of every important fact de nition and theorem from that section that you feel you should memorize 2 Go over your HW assignments and make sure that you understand all of the problems as well as the related problems in the text Pay close attention to the ones you missed the rst time around 3 Think about the questions listed below after you have completed your review Try and gure them out without referring to your notes Warning some of the following questions are much more vague and open ended than what will be on your exam Consider them food for thought 1 Given two vectors in Rd what is an easy way to parameterize the line connecting them ls the parameterization unique 2 Given three vectors in Rd write a parameterization of the plane containing them If the vectors are in R3 what is a way to get an equation of the plane 3 What is the relationship between dot products and length What is your professor s favorite theorem involving dot products 4 Given cylindrical coordinates r 0 2 what gures do you get by holding one ofthe variables constant and letting the other two vary 5 Given spherical coordinates p 0 gt what gures do you get by holding one of the variables constant and letting the other two vary 6 Let f be a real valued function and give the general de nition of a level curve of f If f R2 a R what does a level curve signify on the graph Compare and contrast to a section of the graph 7 Suppose the level curves of a function f R2 a R are horizontal lines If the section in the zz plane is the line 2 ix what is the graph of f ls it possible for the section in the zz plane to be the cubic z 7mg What would the graph look like 8 Consider the sphere of radius 5 in R3 centered at the origin Let s think about it in several different ways First write an equation for the sphere Then give a function f R3 a R that has this sphere as a level surface Third give a function g R2 a R who s graph is the top half of the sphere Fourth give the equation of the sphere in spherical coordinates 9 ls there a function f R2 a R such that as you approach the origin along the y axis the limit is 1 but as you approach the origin along the line y 2x the limit is 2 10 For a graph 2 fm y relate the concept of the partial derivatives fmz0 yo and fyz0 yo to the concept of sections 11 Use fmz0 yo and fyz0 yo to form vectors in R3 that when based at the point mo yo fm0 340 lie in the tangent plane to the graph of f at 0 yo Find the normal vector to the tangent plane using these vectors then verify the formula for the tangent plane on page 133 12 Explain how the derivative matrix gives a linear approximation for a function f R a R Relate your answer to the one dimensional version 13 Technically speaking what is the difference between a path and a curve 14 Let c 071 a R3 be a path True or false If the curve traced by G contains a cusp7 then ct does not exist at that point 15 Suppose f R2 a R and c R a R2 Relate the graph of f o c to the graphs of f and c What is the chain rule for f o c 16 Let fzyz 2 y2 22 and let 9 R3 a R3 take spherical coordinates p0 gt to cartesian coordinates z7 342 a Use the chain rule for Dh Df o g b Compute 87 using another version of the chain rule p 17 Describe at least two different connections between partial derivatives and directional derivatives 18 Suppose f R2 a R a Prove that the gradient is the direction of fastest increase b Prove that the gradient is perpendicular to level curves 19 Suppose f is a real valued function What does 1 is of class Ck mean What does this tell you about the mixed partials of f 20 True or false The mixed partials of a real valued function can exist but not be equal Give an argument for your answer if true7 or an example to the contrary if false 21 State the rst and second order Taylor formulas for a function f R2 a R What is the signi cance of the remainder term in each case 22 If you drop the remainder term in the Taylor formulas7 you get approximations that are linear and quadratic Use a picture to explain the signi cance of this 23 Suppose f is a real valued7 differentiable function Prove that the local extrema must occur at critical points 24 Explain why the second order Taylor formula produces the second derivative test found on page 216 25 Explain why the method of Lagrange Multipliers might more aptly be named the method of parallel gradients 26 Find local and global extrema of the function m7 y y2 cosz on the square 77r2 x 7r2771 y g 1 Use two different methods to nd the extrema on the boundary
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