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# Class Note for MATH 2433 at UH

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COURSE
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TYPE
Class Notes
PAGES
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WORDS
KARMA
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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15
1 Applications 1 Volume lf fXy 20 on then V solid cylinder that has the surface Z fX y as its top sides 113 Area ldXdyarea of Q 111 Mass of a Plate If the density of a plate at a point Xy in the closed bounded region is given by a continuous function X then the mass of the plate is MA wwww iv Center of Mass of a Plate Let the continuous function X be the density function of a plate Then the coordinates XM yM of the center of mass of the plate are given by fX y dXdy is the volume of the Q as its base and vertical XM dXdy yM Where M is the mass of the plate ywwww LLTTLL ll TRIPLE INTEGRALS a De nition Let f fXy Z be continuous on the box T a1 X az b1 y bz 012 02 Let 73 be a partition of T and let 117 and lm be the minimum and maximum values of f on the jjk subebox Rigk Then n m l 1 Lower sum LAP mm Xi yj Zk i1j1k1 m l 11 Upper sum Uf73 2 Mm Xi y Zk i1j1k1 n m l 111 Riemann sum SAP fX y fk Xi y Zk Where 4922 is 111 k1 a point in Rigk The t ple integral of f over T is the unique number I that satis es LAP S S Uf73 Notation I fXyZdXdde T Let for all partitions 73 be an arbitrary closed bounded region in space Then fXyZdXdyZAFXyZdXdde Where T is a box that contains on T 7 and FXyz fXyZ on and FXy 0 b Repeated Integrals If the region is given by agxg b 1ltXgt y zoo my 2 my Type I region then b 152w 2y fXy Z dXdde fX y Z dz 1de Q a 1w INN Note There are ve more types of special regions c Cylindrical Coordinates QfXyZdXddeAfTCOS rsin Zrdrd dz CHAPTER 17 LINE INTEGRALS Given a vector eld FXy PXyi QXyj and a smooth or piecewise smooth curve C C X Xu y yu a g u g b parametric form C ru Xu i yuj a g u g b vector form Or in three dimensions a vector eld FX y Z PXy Z i QX y Zj RXy Z bfk and a smooth or piecewise smooth curve C C X Xu y yu Z 2a a g u g b parametric form C ru Xu i yuj Zu k a g u g b vector form DEFINITION The line integral of F over C is the number given by b Fra r Frur udu C a Alternative notations Fr a r PXy dX QXy dy FT d5 2 dimensions 0 C C 01quot Fr dr PXy Z dX QX y Z dy RXy Z dz FT d5 3 dimensions 0 C C Where FT is the component of F on the unit tangent vector T THEOREM Line integrals are invariant under orientationepreserving changes of parameter THEOREM Reversing the orientation of C changes the sign of the integral LCFra r7CFrdr cf fX 1X 7 fX dX FUNDAMENTAL THEOREM OF LINE INTEGRALS Given a curve C ru a g u g b and a vector eld F If F Vf for some function fX then C Fr a r fB 7 fA Where A ra and B rb DEFINITION The curve C is closed if ra rb COROLLARY 1 If F is the gradient of some function f and the curve C is closed then CFrdrO COROLLARY 2 Independence of Path If F is the gradient of some function f and if C1 and C2 are any two curves Which begin at A and end at B then Frdr Frdr 01 02 DEFINITION The curve C is a simple closed curve if it is closed and rt1 7 rt2 for all a lt Q lt Q lt b The positive direction on C is counterclockwise A simple closed curve is also called a Jordan curve The region enclosed by a simple closed curve is called a Jordan region GREEN S THEOREM Given a simple closed curve C oriented in the counterclockwise direction and a vector eld FXy PXy i QXyj pxydxoxydyQjof p 7 dXd yl y Where is the Jordan region enclosed by C COROLLARY TO GREEN S THEOREM If C is a simple closed curve and if j g that is if F PXy i QXyj is a gradient then y PXydXQXdy0 AREA OF USING GREEN S THEOREM C a simple closed curve enclosing the region Area of ldXdyf idef Xdy ideXdy o c c 2 c SUMMARY CHAPTERS 16 and 17 CHAPTER 16 MULTIPLE INTEGRALS I DOUBLE INTEGRALS a De nition Let f fXy be continuous on the rectangle R a g X g b c g y g 1 Let 73 be a partition of R and let m and IVA7 be the minimum and maximum values of f on the Lj suberectangle R Then M3 M3 1 Lower sum Lf73 ml Xi M 1 1739 M3 M3 113 Upper sum Uf73 Mj Xi M 1 W Z fXy X1 yj Where ngg is a point in R j1 13911 Riemann sum Sf73 l H The double integral of f over R is the unique number I that satis es Lf73 g I g Uf73 for all partitions 73 Notation I fXydXdy R Let be an arbitrary closed bounded region in the plane Then 9 fXydXdyRFXy dXdy Where R is a rectangle that contains and F X fX y on and F X y 0 on R7 b Repeated Integrals If the region is given by a g X b 1X g yg 2X Type I region then b 152m fXy dXdy fXy 1de 9 a 151m If the region is given by c g yg d 10 g X 20 Type II region then 11 2y fXy dXdy fXy WOW 9 0 19 c Polar Coordinates fXydXdy fr cos rsin rdrd r2 r2

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