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# Calc & Analyt Geom IV (HONORS) MATH 2443

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This 6 page Class Notes was uploaded by Einar Nitzsche on Monday October 26, 2015. The Class Notes belongs to MATH 2443 at University of Oklahoma taught by Noel Brady in Fall. Since its upload, it has received 14 views. For similar materials see /class/229291/math-2443-university-of-oklahoma in Mathematics (M) at University of Oklahoma.

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Date Created: 10/26/15

Math 2443002 Calculus IV Spring 2000 Surface Integrals We begin with a discussion of surface area for parametric surfaces Auraquot quotAuru I l f kx 39 Z Fv s ru A I x f ru0 v0quot k v row we quotWm M w Av 3 ru0v 1 i FWWO 110110 Au V y The parametric map ruv takes a rectangle of side lengths Au and A1 to a curvilinear patch on the parametric surface in 3 d as shown in the diagram The area of this patch is approximated up to rst order by the area of a parallelogram in 3 d with sides given by the tangent vectors Auru and Avrv This area is just er lrvl sin QAuAv which is equal to by Calculus Ill 7 cross products lru gtlt rleuAv Taking Riemann sums gives us a surface area integral lru gtlt rvldudv D where D is a region in the uv plane We show how to compute this as a double integral on the next page7 and we de ne surface integrals over a parametric surface for functions or vector elds Surface Area Let S be a parametric surface given by the vector equation ruv ltuv yuv zuvgt where um belong to a region D in the uv plane Then the surface area of S is computed as AreaS ffD lru gtlt rvl dude MD HM deudv 3uu7 3uu7 3u u ND dud Note that when the surface is just the graph of z fy the expression above reduces to D i1 fl fy2dzdy which is a nice generalization of the formula f 4 1 f x2dz for arclength along the graph of y f 96 Surface Integrals One can de ne surface integrals of a function or a vector eld as follows If fyz is a continuous function de ned on a region of R3 which contains the surface S then we de ne fds fruv lru gtlt rvldudv S D If F ltPQRgt is a continuous vector eld de ned on a region of R3 which contains the surface S then we de ne fstds ffsltF d8 ffD Fru1 ru gtlt rv dude Note that the last term above is just an iterated integral There7s quite a bit of computation here and it takes a bit of practice to get pro cient at converting a surface integral into a standard iterated double integral in u and 1 Remark When the surface is just the graph of z fzy my 6 D then we get SFds DF39ltfmrfy71gtd9 dy DltR7nyemgtdxdy Flux When F is the velocity vector eld of a uid ow then 5 F ds represents the net uid owing through the surface S per unit time It is called the ucc ofF across S Math 2443002 Calculus IV Spring 2000 Surface Integrals We begin with a discussion of surface area for parametric surfaces Auraquot quotAuru I l f kx 39 Z Fv s ru A I x f ru0 v0quot k v row we quotWm M w Av 3 ru0v 1 i FWWO 110110 Au V y The parametric map ruv takes a rectangle of side lengths Au and A1 to a curvilinear patch on the parametric surface in 3 d as shown in the diagram The area of this patch is approximated up to rst order by the area of a parallelogram in 3 d with sides given by the tangent vectors Auru and Avrv This area is just er lrvl sin QAuAv which is equal to by Calculus Ill 7 cross products lru gtlt rleuAv Taking Riemann sums gives us a surface area integral lru gtlt rvldudv D where D is a region in the uv plane We show how to compute this as a double integral on the next page7 and we de ne surface integrals over a parametric surface for functions or vector elds Surface Area Let S be a parametric surface given by the vector equation ruv ltuv yuv zuvgt where um belong to a region D in the uv plane Then the surface area of S is computed as AreaS ffD lru gtlt rvl dude MD HM deudv 3uu7 3uu7 3u u ND dud Note that when the surface is just the graph of z fy the expression above reduces to D i1 fl fy2dzdy which is a nice generalization of the formula f 4 1 f x2dz for arclength along the graph of y f 96 Surface Integrals One can de ne surface integrals of a function or a vector eld as follows If fyz is a continuous function de ned on a region of R3 which contains the surface S then we de ne fds fruv lru gtlt rvldudv S D If F ltPQRgt is a continuous vector eld de ned on a region of R3 which contains the surface S then we de ne fstds ffsltF d8 ffD Fru1 ru gtlt rv dude Note that the last term above is just an iterated integral There7s quite a bit of computation here and it takes a bit of practice to get pro cient at converting a surface integral into a standard iterated double integral in u and 1 Remark When the surface is just the graph of z fzy my 6 D then we get SFds DF39ltfmrfy71gtd9 dy DltR7nyemgtdxdy Flux When F is the velocity vector eld of a uid ow then 5 F ds represents the net uid owing through the surface S per unit time It is called the ucc ofF across S Math 24437002 Div Grad and Curl Spring 2000 Nabla The vector differential operator V is pronounced nabld7 and is de ned to be r 8 8 39 j k VEl 8y 82 Grad V can act on a function scalar eld 1 to give a vector eld called the gradient of f 91 91 19f 19f 19f 9f k Vf 8ml8yJ82 lt8m78y782gt Recall that Vf is always perpendicular to the level surfaces of 1 points in the direction of maximum rate of increase of f and that lV is this maximum rate of increase Div The operator V can act on a vector eld F in the same way as a dot product This gives us a scalar eld or function called the divergence of F which is de ned by 7 71313 862 8B dZUF7VF78 yg where F P762713 Curl The operator V can act on a vector eld F in the same was as a cross product This gives another vector eld called the curl of F which is de ned by i j 12 cmlFVxF ltRy Q27Pz RvawVPZgt P Q R Laplace operator The Laplace operator acts on a scalar eld or function and is de ned to be div gmd 821 821 821 2 7 Vf78x28y2822 Two interpretations of Green7s Theorem The two equations below are versions of Green s theorem They help us gain an intuitive understanding of div and curl ds Fdr curlF KdA Tangential components7 C C D ds divFdA Normal components C D Stokes Theorem Let S be an oriented7 piecewisesmooth surface bounded by a positively oriented7 simple7 closed7 piecewisesmooth curve C Let F be a vector eld Whose components have continuous partial derivatives on an open region of R3 which contains S Then CFdr ScurlFds Divergence Theorem Let E be a simple solid region in R3 Whose boundary surface S has a positive outward orientation Let F be a vector eld Whose components have continuous partial derivatives on an open region of R3 which contains E Then SFds EdwFdv Physical Interpretations Let F be the velocity vector eld of a uid ow Let P0 0340213 be a point in the uid ow7 and n be a unit vector based at P0 Then curlFP0 n measures the rotating effect of the uid about the axis n This rotating effect is greatest about an axis which points in the direction of curlFP0 Also divFP0 measures the net rate of outward ux per unit volume at P0

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