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Multivariable Calculus

by: Rae Kutch

Multivariable Calculus MATH 251

Rae Kutch
GPA 3.8


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Class Notes
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This 5 page Class Notes was uploaded by Rae Kutch on Saturday September 12, 2015. The Class Notes belongs to MATH 251 at West Virginia University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/202665/math-251-west-virginia-university in Mathematics (M) at West Virginia University.

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Date Created: 09/12/15
Review for Chapter 13 1 Given a parametric planar curve z t7 y 9257 where a g t b7 how to eliminate the parameter Use substitutions or use trigonom How to parametrize a curve fzy 0 For polar form r 1 09 one can set Mt ft cost and yt ft sint for the special form y m one can set Mt tand yt ft etry indentities7 etc 3 2 Differentiation of parametric curves and the slope i dydt dzy dm 7 elmalt7 dmz For polar coordinates7 dy 7 r sin0 rcos0 dz T cost97rsint97 where 1 denotes the angle between the tangent line llMt 7 dmdt39 m r 107 at P and the radius OP from the origin cot gt 3 Review of integration in Chapter 6 and how to apply the formulas to paramatric curves Area under curve Vol of rev around X axis Vol of rev around y axis Arc length Area of surf of rev around X axis Area of surf of rev around y axis 4 Operations of vectors The addition and the multiplication of vectors operations of numbers with some exceptions see 7 3 on page 643 The following are some reminders Afydm v fvryzdz Vy fmzdy sifdsf dm2dy Am leylds Ay lemlds behave pretty much the same way as the on page 6337 9 on page 634 and Theorem i The dot product of two vectors outputs a number ii The scalar product and the vector product output vectors iii The vector product of two vectors is not commutative 5 Some important facts 1 ii 1v 3 x I is perpendicular to both 3 and b 6 cos 0 vi x El W11 sin0 where 9 in v and vi AAAAAA is the angle between 3 and I lf 3 7 6 then 73 Jld l is the unit vector that has the same direction of Z Perpendicular test 67 0 iff Z and b are perpendicular in Parallel test 3 x b 0 iff Z and b are parallel Review for Chapter 14 1 Operation of vecters The addition and the multiplication of vectors behave pretty much the same way as the operations of numbers with some exceptions see 7 of page 633 9 of page 634 and Theorem 3 of page 643 The following are some reminders i The dot product of two vectors outputs a number ii The scalar product and the vector product output vectors iii The vector product of two vectors is not commutative 2 The distance formula and its relation to dot products If 771188 its head in 1 y121 and tail at 2 y222 then Flt 1 7 2341 7 yz 21 7 22 gt and 1712 7 7 951 7 2 241 7 y22 21 7 222 3 Important facts the angle 9 below is the angle between the two vectors 3 and i Z 01 cos 0 and so i and hare perpendicular if and only if d o 1 0 ii J x 32 32ng sin2 0 and so 3 and I are parallel to each other if and only if 3 x I 6 the zero vector iii 3 x I is perpendicular to both i and 4 Sction 141 Direction angles and numbers Let F lt myz gt Let the angles between F and the z y and z axes be a B and 39y respectively Then these angles are the direction angles of F and the direction numbers are 7 a 3 a I cos T13 cosy T W W 5 Vector components Section 141 The component of 3 along I is L 7 COSOz 2 W7 L a b Compgd a 151 6 Equations of a line Section 143 Let L be a line in space that is parallel to 7 lt a b c gt and passes through 0340213 Then the parametric equation of L is rt lt at m0bt 1 yo ct 20 gt and the symmetric equations of L is 957900 y7yo 272 0 a b c 39 7 Facts about planes Section 143 The plane passes 0340213 with normal vector 7 lt a bc gt has equation ltabcgt0ltz7x0y7y0z7zo gtO The angle between two planes is the angle between the two normal vectors 8 Basic facts on vector functions Section 144 Vector functions and their limits deriva tives and antiderivatives One key thing to remember do each of these componentwise Note that the product rules for derivatives are very similar to the product rule for ordinary functions with exceptions in vector products 9 Basic techniques in motions Section 145 Given r65 nd the velocity the speed the acceleration the unit tangent vector the principal unit normal vector the tangent component and the normal component of the acceleration and the curvature See your notes for routines Conversely given the velocity and the acceleration together with some initial conditions nd the position vector rt See your notes for routines 10 Basic techniques of studying graphs of equations Section 146 Given the equation of a surface and the cuttling planes one can describe the the graph by using traces layers and cylinders according to the nature of the given surface One can also use the traces layers andor cylinders to scketch the graph 11 A technique of obtaining equations from descriptions Section 146 Given the equation of a plane curve C and an axis L nd an equation of the surface generated by revolving C about L by using distance formula 12 Section 147 The Cylindrical and the spherical coordinates The fomulae about cylindrical coordinates zrcos0 y yrsin0 andz2y2r2tan0 22 The fomulae about spherical coordinates x p cos 9 sin 1 y psin0sin gt and 2 y2 22 p2 z p cos 1 Review for Chapter 16 1 Evaluation of double integrals Ly coordinates 17 92W fxydA fydydx7 ifR is a S x S b7 91z S y S gg R a 91W 1 h2 fltzygtdA mmdwz ifRiscsysd h1ltygtxsh2y R C hill When you can evaluate the integral by either way7 you may want to choose a simpler way 2 Evaluation of double integrals Polar coordinates b T fxydA g f7 cost9rsint97quotdt9dr7 if R is a S r S b7 91r S 6 S 927 R a 917 b h 9 Rfac7ydA 49 frcos0rsin0rdrd6 ifR is a g 9 g b h16S r 3 1120 1 1 A useful fact dA rdrd dxdy 2 Some applications of double integrals 2a Area of region R is f R dA 2b The volume beween z fy and z g7y when Ly are in R is f fRfz7y 7 HOWA 2c Applications in Physics Let pxy be the density of larnina whose region is R Then the mass and the centroid of the lamina 2 is mass m ffR pz7ydA iffRzJWzWA J JR WWWWA 3 Evaluation of triple integrals rectangular coordinates The main idea is the Rl Ql same as the cross section idea The following gives a way to reduce a triple integral to a double integral TfyzdV Rhhlmy fyzdzdA if T is h1zy S 2 S hgy7 Ly in R 2 mill 4 Evaluation of triple integrals cylindrical coordinates and spherical coordi nates The main relationship among rectangular7 cylindrical and spherical coordinates is dV dxdydz rdrd dz p2 sin d d6dp 5 Some applications of triple integrals 5a The volume of the solid T is dV 5b The mass of a solid T with density pxyz is pyzdV 5c The centroid of T with density pxyz is iii where TAMQMMWV Aypz7y7zdViAZPvWdV 6 Use double integral to nd the surface area If a curface is given by ruv lt xuv7yuvzuv gt7 where um is in R7 then the area of the curface is Ri gtlt gldfl


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