Study Guide for MATH 223 at UA
Popular in Course
Popular in Department
This 4 page Study Guide was uploaded by an elite notetaker on Friday February 6, 2015. The Study Guide belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 28 views.
Reviews for Study Guide for MATH 223 at UA
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 02/06/15
Final Exam Math 223 Study Guide Spring 2010 Dot Products and Cross Products v 7 vlr1 vzr2 v3r3 HTIHHFHCOSQ i 2 IE 7 X7 v1 v2 v3 HvHHrHcos r1 r2 3 Local Linearity For every graph fXy approximately equals LXy for some local linear function The graph 2 LXy is the tangent plane to fXy at the point ab Z fxaabx a fyaby b faab tangent planes The differential is defined below df fxxydx fyabdy the differential Gradients and the Directional Derivative The directional derivative gives the slope of fXy in the direction of a fa hulb hu2 fab fgab11 3 h fxaba1fyabu2 The gradient is de ned as the vector function below gradf Vf ii ij 9x y 91 The following are aspects of the gradient 1 gradf Jim 130117 2 Mala gradfab o a Ch ain Rule For simple functions the following applies If zXy amp Xt amp yt then Local Extrema Let Vf gradf 0 then nd points i fa f If Vfx0y0 0 let D W fufyy fw2 fyx fyy If D gt 0 f gt 0 local minimum f lt 0 local maximum D lt 0 saddle point Integration 12 320 d My fffxygtdA f ffltxydydx f ffltxydxdy R u 310 0 My 1397 Maw20w fxyzdV f f ffxyzdzdydx u MOE 141001 Cylindrical Coordinates xRcos0 yRsin0 zz dVRdzde0 Polar Coordinates xpsin cos0 ypsin sin0 z pcos dV p2 sin dpd0d Line Integrals fig lim SFQyAZ C i0 limbo 7 f F d7 fF7t 397 tdt C u Fundamental Theorem fcgradf d7 fQ fP A vector field F is said to be path independent or conservative if the line integral has the same value along any piecewise smooth path C from P to Q lying in the domain of F Gradient Fields If F is a continuous pathindependent vector eld on an open region R then F grad f for some f defined on R f is a potential function of F A vector field is pathindependent if and only if the line integral is zero for every closed curve C If curlF LFI 0 then F is a gradient field Curl Test 926 y Green s Theorem IF 39 d7 f LFZ LFI dxdy Note Curve must be oriented CCW C R 6x 9 Flux Integrals ffdAfF dA11m PM 539 S HMHrO Flux through a Graph 2 fgx y fF dA fFxy fx y in fy j kdxdy S R Flux through a Cylinder fF dA f FR 0z cos 9 sin 0Rdzd0 S T Flux through a Sphere f dAfFdA S S Theorems Divergence Geometric De nition of Divergence d a 1 SF dA ivF x z imi y V lgt0 VOLS Cartesian Coordinate De nition of Divergence 9F 9F 9F divF 71 72 73 9x y 91 Divergence Theorem f d3 fdiv dv S W Curl Circulation Density CirculationinC fCF 39 d7 ctrc FOc yz 1m Area Areueo AreaC De nition of Curl curz wx 2 y 91 91 9x 9x 9y Stokes Theorem d7 fourl dg C s
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'