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# Math Study g MA 261

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This 19 page Study Guide was uploaded by Nikki Bee on Wednesday June 10, 2015. The Study Guide belongs to MA 261 at Purdue University taught by in Spring 2015. Since its upload, it has received 43 views. For similar materials see Calculus 3 in Mathematics (M) at Purdue University.

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Date Created: 06/10/15

MA 261 Spring 2011 Study Guide 1 1 Vectors in R2 and R3 a V ltabc ai l 93 CE vector addition and subtraction geometrically using paral lelograms spanned by 11 and 7 length or magnitude of v ltabc WI 2 Va2 b2 c2 directed vector from P0a0y0zo to P1a1y1zl given by v P0P1 P1 P0 331 3307M 240721 20gt b Dot or inner product of 5 a1a2a3 and l 191192193 5 b albl a2b2 a3b3 properties of dot product useful identity 5 5 52 angle between two vectors 5 and b 39 b cos6 2 lm 5 J b if and only if 5 b 0 the vector in R2 With length 7 With angle 6 is a v ltrcos67 sin6 y r 9 gtlt 9 pro b mmab a d Cross product only for vectors in R3 E a a a a a a a X b al a2 a3 2 3 i 1 3 j 1 2 k b2 b3 b1 b3 b1 b2 b1 b2 b3 properties of cross products 5 X l is perpendicular orthogonal or normal to both 5 and 6 area of parallelogram spanned by 5 and l is A gtlt 7 gt b the area of the triangle spanned is A gtlt 7 gt b 2 Equation of a line L through P010 yo 20 with direction vector d a b c Vector Form Ft 2 130310 20 td 1 30 I at Parametric Form y 2 yo I bt z 20 ct 33 0y y0Z Zo Symmetric Form a b c gt d39 xoy0zo 170 2 20 If say I 07 then a a C 7y2y039 Equation of the plane through the point P010 yo 20 and perpendicular to the vector a b c is a normal vector to the plane is 330 y yo7 z 20 0 Sketching planes consider 51334 2 intercepts gt Quadric surfaces can sketch them by considering various traces7 ie7 curves resulting from the intersection of the surface With planes 1 k y k and or z k some generic equations have 4 the form 172 2 Z2 a Ellzpsozd 7 g 1 b Elliptic Paraboloz39d E 2 31 2 c a2 b2 c Hyperbolic Paraboloz39d Saddle E x y Z2 2 312 d Cone 0 2 g 9 2 2 312 Z2 e Hyperbolozd of One Sheet g 9 2 g 1 2 312 Z2 f Hyperbolozd of Two Sheets 1 10 11 12 t 2 position of a particle F t v t velocity 575 2 Vector valued functions Ht 2 lt f t7 gt ht tangent vector F t for smooth curves7 unit tan Wt lf t differentiation rules for vector T t unit normal vector Nt gent vector IT 75 functions including i gbt 905 gbt v t gb t v t Where gbt is a real valued function ii iii quot7 v v xv gtltv gtltv WWW Wt 7 t7 iv Where gbt is a real valued function Integrals of vector functions Ht dt lt f t dt7 gt dt7 ht dtgt arc length of curve b t parameterized by t is L lf dt arc length function 8t lf du reparameterize by arc length 68 2 Ft Where 758 is the inverse of the arac length function 8t the lTm oe39 d2 oz If t39 ampf 39 quot curvature of a curve parameterized by t is K v t F t acceleration Iquot t 7t 2 speed Newton s 2quot Law F 2 m5 Domain and range of a function f1y and f1y z level curves or contour curves of f 13 y are the curves f 13 y k using level curves to sketch surfaces level surfaces of f 13 y z are the surfaces f1y z k fw hyy f937y 5 Partial derivatives fa y 2 1205 y Illinti 91 h 7 3f fw7yh fwy 82f 8y 3334 fy1 y Illin h higher order derivatives fmy ay 83 32f 32f fyy 8 242 fym m etc mixed partials Equation of the tangent plane to the graph of z f 13 y at 130 yo zo is given by Z 20 fm07y0x 0 fy3307y0y 30 Total differential for z f 13 y is dz 2 df d3 dy total differential for w f 13 y z 33 y is dw 2 df d3 g dy g dz linear approximation for z f 13 y is given by Az dz7 833 y dz 3f ie7 f1 Aa y l Ay fay gdx l gi ydy7 Where A3 d513 Ag 2 dy Linearization of f1y at db is given by Lay fdb 120017 b1 a fydby b Hazy ay near cub 13 Different forms of the Chain Rule Form 17 Form 2 General Form Tree diagrams For example 9593 dx a Ifz fay and 2422405 then dt an dt 8y dt zfxy 5 13 8 5x 8y x y dx d dt y dt 1 t b Hz 2 f1y and then and 88 83 698 g 88 t 5 8t zfxy E at VW axax ay ay askt 9E s t S t etc 83875 14 Implicit Differentiation and Directional Derivative Implicit Differentiation d Part I If Fxy 0 de nes y as function of 1 ie7 y then to compute d y a d differentiate both sides of the equation F 13 y 0 Wrt 1 and solve for a 8 If Fayz 0 de nes 2 as function of 1 and y ie z then to compute 8 2 a 8 8 differentiate the equation F 3yz 0 Wrt 1 hold y xed and solve for For 8 2 33 y 8 differentiate the equation F 13 y z 0 Wrt y hold 1 xed and solve for y 8F Part II39 If Fa y 0 de nes y as function of 1 gt 39 7 dx 8 7 831 8F ELF h39l39fF 0d f t39 f d gt 8 x d W1e1 xyz eneszas unc1onoxan y 833 a F an ay 82 82 Directional derivative Directional derivative of f 13 y at 130 yo in the direction 11 D f 5130 yo V f 5130 yo 11 where 11 must be a unit vector MA 261 Spring 2011 Study Guide 1 1 Vectors in R2 and R3 a V ltabc ai l 93 CE vector addition and subtraction geometrically using paral lelograms spanned by 11 and 7 length or magnitude of v ltabc WI 2 Va2 b2 c2 directed vector from P0a0y0zo to P1a1y1zl given by v P0P1 P1 P0 331 3307M 240721 20gt b Dot or inner product of 5 a1a2a3 and l 191192193 5 b albl a2b2 a3b3 properties of dot product useful identity 5 5 52 angle between two vectors 5 and b 39 b cos6 2 lm 5 J b if and only if 5 b 0 the vector in R2 With length 7 With angle 6 is a v ltrcos67 sin6 y r 9 gtlt 9 pro b mmab a d Cross product only for vectors in R3 E a a a a a a a X b al a2 a3 2 3 i 1 3 j 1 2 k b2 b3 b1 b3 b1 b2 b1 b2 b3 properties of cross products 5 X l is perpendicular orthogonal or normal to both 5 and 6 area of parallelogram spanned by 5 and l is A gtlt 7 gt b the area of the triangle spanned is A gtlt 7 gt b 2 Equation of a line L through P010 yo 20 with direction vector d a b c Vector Form Ft 2 130310 20 td 1 30 I at Parametric Form y 2 yo I bt z 20 ct 33 0y y0Z Zo Symmetric Form a b c gt d39 xoy0zo 170 2 20 If say I 07 then a a C 7y2y039 Equation of the plane through the point P010 yo 20 and perpendicular to the vector a b c is a normal vector to the plane is 330 y yo7 z 20 0 Sketching planes consider 51334 2 intercepts gt Quadric surfaces can sketch them by considering various traces7 ie7 curves resulting from the intersection of the surface With planes 1 k y k and or z k some generic equations have 4 the form 172 2 Z2 a Ellzpsozd 7 g 1 b Elliptic Paraboloz39d E 2 31 2 c a2 b2 c Hyperbolic Paraboloz39d Saddle E x y Z2 2 312 d Cone 0 2 g 9 2 2 312 Z2 e Hyperbolozd of One Sheet g 9 2 g 1 2 312 Z2 f Hyperbolozd of Two Sheets 1 10 11 12 t 2 position of a particle F t v t velocity 575 2 Vector valued functions Ht 2 lt f t7 gt ht tangent vector F t for smooth curves7 unit tan Wt lf t differentiation rules for vector T t unit normal vector Nt gent vector IT 75 functions including i gbt 905 gbt v t gb t v t Where gbt is a real valued function ii iii quot7 v v xv gtltv gtltv WWW Wt 7 t7 iv Where gbt is a real valued function Integrals of vector functions Ht dt lt f t dt7 gt dt7 ht dtgt arc length of curve b t parameterized by t is L lf dt arc length function 8t lf du reparameterize by arc length 68 2 Ft Where 758 is the inverse of the arac length function 8t the lTm oe39 d2 oz If t39 ampf 39 quot curvature of a curve parameterized by t is K v t F t acceleration Iquot t 7t 2 speed Newton s 2quot Law F 2 m5 Domain and range of a function f1y and f1y z level curves or contour curves of f 13 y are the curves f 13 y k using level curves to sketch surfaces level surfaces of f 13 y z are the surfaces f1y z k fw hyy f937y 5 Partial derivatives fa y 2 1205 y Illinti 91 h 7 3f fw7yh fwy 82f 8y 3334 fy1 y Illin h higher order derivatives fmy ay 83 32f 32f fyy 8 242 fym m etc mixed partials Equation of the tangent plane to the graph of z f 13 y at 130 yo zo is given by Z 20 fm07y0x 0 fy3307y0y 30 Total differential for z f 13 y is dz 2 df d3 dy total differential for w f 13 y z 33 y is dw 2 df d3 g dy g dz linear approximation for z f 13 y is given by Az dz7 833 y dz 3f ie7 f1 Aa y l Ay fay gdx l gi ydy7 Where A3 d513 Ag 2 dy Linearization of f1y at db is given by Lay fdb 120017 b1 a fydby b Hazy ay near cub 13 Different forms of the Chain Rule Form 17 Form 2 General Form Tree diagrams For example 9593 dx a Ifz fay and 2422405 then dt an dt 8y dt zfxy 5 13 8 5x 8y x y dx d dt y dt 1 t b Hz 2 f1y and then and 88 83 698 g 88 t 5 8t zfxy E at VW axax ay ay askt 9E s t S t etc 83875 14 Implicit Differentiation and Directional Derivative Implicit Differentiation d Part I If Fxy 0 de nes y as function of 1 ie7 y then to compute d y a d differentiate both sides of the equation F 13 y 0 Wrt 1 and solve for a 8 If Fayz 0 de nes 2 as function of 1 and y ie z then to compute 8 2 a 8 8 differentiate the equation F 3yz 0 Wrt 1 hold y xed and solve for For 8 2 33 y 8 differentiate the equation F 13 y z 0 Wrt y hold 1 xed and solve for y 8F Part II39 If Fa y 0 de nes y as function of 1 gt 39 7 dx 8 7 831 8F ELF h39l39fF 0d f t39 f d gt 8 x d W1e1 xyz eneszas unc1onoxan y 833 a F an ay 82 82 Directional derivative Directional derivative of f 13 y at 130 yo in the direction 11 D f 5130 yo V f 5130 yo 11 where 11 must be a unit vector MA 261 Fall 2012 Study Guide 3 You also need Study Guides 1 and 2 for the Final Exam Line integral of a function f 13 y 2 along C parameterized by 1 2 3375 y yt z 2t and a S t S b is C fw7y7z ds bfat ya 20 dt independent of orientation of C other properties and applications of line integrals of f Remarks a 0 fx y 2 ds is sometimes called the line integral of f with respect to arc length b emsmas bfltxlttgtylttgtzlttgtgtx39lttgtdt c Cfwyzdy bfltwlttgtylttgtzlttgtgty39lttgtdt d C mews bfltxlttgtylttgtzlttgtgtz39lttgtdt Line integral of vector eld F a y 2 along C parameterized by t and a S t S b is given by b F df Ff t m dt C 1 depends on orientation of C other properties and applications of line integrals of f Connection between line integral of vector elds and line integral of functions fFdFF Tds C C Where 39f is the unit tangent vector to the curve C If F013 3 Z PC7341 2 iUr 62013731 Zi R613 3 Z 12 then f df PwyzdwQwyzdyRwyzdz C C Work FquotdI quot C FUNDAMENTAL THEOREM OF CALCULUS FOR LINE INTEGRALS Vfdf f Fb f Fa c C rb 8Q8P 6 A vector eld 2 135 y i Qayjis conservative ie F Vf if E 8y how to determine a potential function f if Vf 8Q 8P 7 GREEN S THEOREM Pxy d3 Qxy dy 8 8 dA C boundary of D C D 33 y As a consequence of Green s Theorem one has 1 xdy ydxzf xdyz f ydxAreaD 2 C C C a 7 8 7 8 8 Del Operator 1 a yJ a f E 8 8 8 lFZV F cur x 833 ay 82 P Q R Properties of curl and divergence O 12 if FOE y Z PC7341 2 iUr 62013731 2 R613 3 127 then 8P 8Q 8R and leF VFa xa yg i If curl F 6 then F is a conservative vector eld ie7 Vf ii If curl F 6 then F is irrotational if div F 07 then F is incompressible 9 Parametric surface S Hum ltxuvyuvzuv Where u39v E D V D L L AK X Normal vector to surface S 2 Eu gtlt ECU tangent planes and normal lines to parametric surfaces 10 Surface area of a surface S 11 12 i AS Iquotu gtlt FvdA D ii If S is the graph of z ha y above D then AS 1 Sh3 332 8ho y2 dA D Remark dS lfu gtlt EU dA differential of surface area While dS 17 X EU dA The surface integral of f 13 y 2 over the surface S i fayzgtd8 flt uvgtlr ugtltmdA ii If S is the graph of z ha y above D then dS h aha 2 ME 2dA Ami2 Dfltxy xygtgt 1lt M lt ygt The surface integral of F over the surface S recall dS 171 X EU dA fSFd BFmmm fSF d AF dszf uxmm If S is the graph of z ha y above D With oriented upward and F P Q R then d D P Qg ZR M i Connection between surface integral of a vector eld and a function fgFdsS ds The above gives another way to compute S F ZS ii F dS dS 1 of F across the surface S s s n S 13 STOKES THEOREM F if curl d recall7 curlf V X C S n F dfquot circulation Of F around C C 14 THE DIVERGENCE THEOREMGAUSS THEOREM F d divf dV 5 E recall7 divf V n 34 15 Summary of Line Integrals and Surface Integrals LINE INTEGRALS SURFACE INTEGRALS CFtwherea t b S Fu39v Where u39v E D d8 l dt differential of arc length dS lfu gtlt EU dA differential of surface area d8 length of C 0 dS 2 surface area of S S C m y 2 ds mt lf tl dt independent of orientation of C fS 357972 ff u7v If x m dA independent of normal vector R r m dt ds a x a M fc df fab m m dt depends on orientation of C fS 39dng uw E u gtlt EU dA depends on normal vector LFdFLFTds The circulation of F around C fS dsSF dS The flux of F across S in direction 16 Integration Theorems b FUNDAMENTAL THEOREM OF CALCULUS F 3 d3 Fb Fa b FUNDAMENTAL THEOREM OF CALCULUS FOR LINE INTEGRALS V f df f E b f E a C a rb GREEN S THEOREM D STOKES THEOREM curlF d F df S C DIVERGENCE THEOREM divf dV 2 f F d E s n S 34 MA 261 Spring 2011 Study Guide 2 3 3 f f properties of gradients gradient points in 81 8y direction of maximum rate of increase of f The maximum value of the directional derivative is equal to Vfa0 yo J level curve fa y C and7 in the case of3 variables Vfa0 yo 20 I level surface f1 y z C Gradient vector for fay Vf337y nVfxoyo zo nVfxoyo x fxyc 39lt x0 Vo sz oy Relativelocal extrema critical points V f 6 or V f does not exist 2nd Derivatives Test A critical points is a local min if D fmfyy 3y gt 0 and fm gt 07 local max if D gt 0 and fm lt 07 saddle if D lt 0 absolute extrema Max Min Problems Lagrange Multipliers Extremize f subject to a constraint 92 2 C solve the system V f AVg and 92 2 C m n Double integrals Midpoint Rule for rectangle f f 3334 dA Z Z ay i AA R 21 31 h1y S 33 S h2y 9132 33139200 Type I region D Type II region D C S y S d 7 agxgb 92 11 b iterated integrals over Type I and II regions f 13 y dA f 13 y dy d3 and D 1 9130 d h2y f 51334 dA f 3334 d3 dy respectively Reversing Order of Integration regions D c h1y that are both Type I and Type II properties of double integrals Integral inequalities mA g dA 3 MA Where A 2 area of D and m S g M D on D Change of Variables Formula in Polar Coordinates if D hlw S T S h26 then ozlt6 6 h209 fxydA f7 cos67 rsin6rdrd6 D a 1110 T 6 Applications of double integrals a Area of region D is AD dA D b Volume of solid under graph of z f1 y Where fa y 2 0 is V fa y dA D c Mass of D is m p1y dA Where p1 y 2 density per unit area sometimes write D m dm Where dm 2 p1 y dA D d Moment about the az aXis M9 y p1 y dA moment about the y aXis My 2 1 p1 y dA D D MyDap33ydA Mm f ypx7ydA e Center of mass 5 y Where E 7 g E pct y M D m fDpw7ydf1 Remark centroid 2 center of mass When density is constant this is useful 7 Elementary solids E C R3 of Type 1 Type 2 Type 3 triple integrals over solids E fayzdv fxyzdz dA for E 33731 6 D May S 2 3 W731 VE volume of solid E is dV applications of triple integrals mass of a solid moments E about the coordinate planes Mmy Mm Myz center of mass of a solid 3amp2 8 Cylindrical Coordinates 7362 1 rcos6 From CC to RC y rsin6 22 X Going from RC to CC use 32 y2 r2 and tan6 y make sure 6 is in correct quadrant a 9 Spherical Coordinates p 6 gb Where 0 3 gb 3 7r 1 psin gb cos6 From SC to RC y psin gb sin6 z pcosgb Going from RC to SC use 32 y2 Z2 2 p2 tan6 g and cosgb E a p 10 11 12 1 rcos6 y rsin6 22 ffE xvvadVffEfrcos6rsin6zrdzdrd6 T de39rdzd39rdH Triple integrals in Cylindrical Coordinates 1 psin gb cos6 y psingb sin6 z pcosgb ffEfxyz dVEfpsingbcos6 psingbsinQ pcos p2sjn dpd d9 T dV 2 p2 singb dp dqb d0 Triple integrals in Spherical Coordinates Vector elds 011 R2 and R3 x 3 135 3 62013731 and FOE y Z 135 3 62613731 R013 31 F is a conservative vector eld if F V f for some real valued function f

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