It is recommended to change the limits of integration while doing a substitution.

Transcription

1 MAT 21 eptember 7, 216 Review Indrjit Jn. Generl Tips It is recommended to chnge the limits of integrtion while doing substitution. First write the min formul (eg. centroid, moment of inerti, mss, work flow, etc.) then put the limits of the integrtion. rwing picture is recommended. Pictures of the curve/re/solid will help you to choose the right coordinte system nd to find the limits of the integrtion. Observe the symmetry of both the solid nd the density before ttempting to find the centroid. ymmetry elimintes unnecessry clcultions nd helps to choose the right coordinte system. Alwys write the sign on top of vector quntity. Note tht d x represents the prtil derivtive with respect to x, nd respect to x. These nottions should be used properly. dx represents the derivtive with 1. hpter 15 ouble Integrtion rw the region of integrtion. This will help you to find the limits of integrtion ccurtely. If you hve b g2(y) xg 1(y) f(x, y) dxdy, figure out the limits of x first (by holding y t generic position), then figure out the limits of y. For dydx, it is other wy round. If the integrtion seems impossible to compute, try reversing the order of integrtion or try converting it to the polr coordinte system. There is lwys n extr r fctor in the integrtion in polr coordinte system, nmely r drdθ. Triple Integrtion: If you hve b g2(x) yg 1(x) h2(x,y) zh 1(x,y) f(x, y, z) dzdydx, figure out the limits of z first (by holding x, y t generic position). Then tke the shdow of the solid onto the xy-plne nd figure out the limits of y nd then limits of x. If the integrtion is in different order eg. dxdydz, then do x first, tke the shdow onto yz-plne nd figure out the limits of y nd z. If the solid is cylindriclly symmetric i.e., the bse of the solid is prt of circle or n ellipse, often it is convenient to use the cylindricl coordinte system. For exmple, consider the solid () bounded by the cylinder y 2 + z 2 9, the yz-plne, nd the prboloid x y 2 + z 2. This solid is bsed on the yz-plne nd sticks out of yz-plne towrds the positive x xis. It is better to use dx rdrdθ order of integrtion i.e., f(x, y, z) dv will become 2π 3 r2 f(x, r cos θ, r sin θ) dx rdrdθ. (1) 1

2 Reltions between the rectngulr, cylindricl, nd the sphericl coordinte system: If (x, y, z) is the coordinte of point P in the rectngulr coordinte system, (r, θ, z) is the coordinte of the sme point in the cylindricl system, nd (ρ, φ, θ) in the sphericl system then x r cos θ y r sin θ x ρ sin φ cos θ y ρ sin φ sin θ z z z ρ cos φ. The cylindricl coordintes re not lwys represented by (r, θ, z). ometimes, depending on the orienttion of the solid, (x, r, θ) or (r, y, θ) re convenient. For exmple, (x, r, θ) ws used in (1). imilr rgument is true for sphericl coordinte system too. There is lwys extr r fctor in the integrtion in cylindricl coordinte system (i.e., r dzdrdθ), nd extr ρ 2 sin φ in the sphericl coordinte system (i.e., ρ 2 sin φ dρdφdθ). If the solid is prt of sphere or n ellipsoid, then it is better to use the sphericl coordinte system. For exmple, consider the solid () trpped inside the sphere x 2 + y 2 + z 2 7 nd outside of the cone z ± x 2 + y 2. Then f(x, y, z) dv becomes 2π 3π/4 7 π/4 f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)ρ 2 sin φ dρdφdθ. Mss, entroid, Moment of Inerti: Let δ(x, y, z) be the density of solid. Mss M δ(x, y, z) dv, where dv dzdydx dz rdrdθ ρ2 sin φ dρdφdθ depending on the chosen coordinte system. First moments: M yz xδ(x, y, z) dv M xz entroid: ( x, ȳ, z) (M yz /M, M xz /M, M xy /M). yδ(x, y, z) dv M xy zδ(x, y, z) dv. Moment of Inerti: Let L be the xis of rottion, nd d(x, y, z) be the distnce of generic point (x, y, z) from L. Then the moment of inerti of the solid with respect to the line L is I L d(x, y, z) 2 δ(x, y, z) dv. (2) For exmple, if L is the x xis, then (2) becomes I x (y 2 + z 2 )δ(x, y, z) dv. If L is the line prllel to z-xis nd pssing through the point (1, 2, 1) then (2) becomes [ I L (x 1) 2 + (y 2) 2] δ(x, y, z) dv. hnge of Vribles: In multiple integrls, sometimes mking chnge of vribles mkes things lot esier. 2

3 Let f(x, y, z) dx dy dz be triple integrl of function over region R, where R is solid in the R xyz spce. We mke chnge of vrible vi x g 1 (u, v, w) y g 2 (u, v, w) z g 3 (u, v, w). After mking this chnge R becomes region G in the uvw-spce. The Jcobin of the chnge of vrible is given by the determinnt x x x (x, y, z) (u, v, w) u v w u v w. Note tht the old vribles re differentited with respect to the new vribles. u v w The integrl f(x, y, z) dx dy dz will be trnsformed s follows R f(g 1 (u, v, w), g 2 (u, v, w), g 3 (u, v, w)) (x, y, z) (u, v, w) G dw dv du. Notice tht the bsolute vlue of the Jcobin (x,y,z) (u,v,w) Few importnt exmples of chnge of vribles is tken. () R : x 2 / 2 + y 2 /b 2 1 on xy coordinte system. hnge of vribles; x r cos θ, y br sin θ. New region G : r 1 is the unit circle. Jcobin (x,y) (r,θ) br. (b) R : x 2 / 2 +y 2 /b 2 +z 2 /c 2 1 on xyz coordinte system. hnge of vribles; x ρ sin φ cos θ, y bρ sin φ sin θ, z cρ cos φ. New region G : ρ 1 is the unit sphere. Jcobin (x,y,z) (ρ,,φ,θ) bcρ 2 sin φ. 2. hpter 13 Here is the min frmework urve: r(t) x(t) î + y(t) ĵ + z(t) k. Velocity: v(t) d r dt x (t) î + y (t) ĵ + z (t) k. peed: v(t) x (t) 2 + y (t) 2 + z (t) 2. Unit tngent vector: T (t) v(t) v(t). Arc length prmeter: s(t) t t v(τ) dτ, where t is the initil time nd t is the finl time. ds dt v(t). Arc length: b v(t) dt, where t is the initil time nd t b is the finl time. urvture: κ d T /dt v(t). Unit norml vector: N d T /dt d T /dt. 3

4 Binorml vector: B T N. Accelertion: d v dt. Tngentil nd norml components of ccelertion: T T + N N, where the tngentil component is T d v dt nd the norml component is N κ v T. 3. hpter 16 There re minly two types of line integrls, Line integrl of sclr field nd line integrl in vector field. (1) Line integrl of sclr function: This is used to compute the re bounded bove curve nd below surfce (see figure 16.5). This cn lso be used to compute the length, mss, centroid, nd moment of inerti of wire. Let r(t) x(t)î + y(t)ĵ + z(t)k; t b be the eqution of spce curve, nd f(x, y, z) be sclr function then the line integrl of f(x, y, z) over the curve is given by f(x(t), y(t), z(t)) v(t) dt. (3) If r(t) is plner curve (eg. r(t) y(t)ĵ + z(t)k is plner curve on the yz-plne), nd f is function defined on the sme plne (eg. f(y, z) is function defined on the yz-plne), then (3) gives the re of the wll bounded in between the curve nd the function f (similr to figure 16.5). If f(x, y, z) δ(x, y, z) is the density of the wire, the (3) gives the mss of the wire. If f(x, y, z) (x 2 + y 2 )δ(x, y, z), where δ(x, y, z) is the density of the wire, then (3) gives the moment of inerti of the wire with respect to the z-xis etc. ee the Tble 16.1 for complete set of formul nd observe the similrity with the formul given in pge 2. (2) Line integrl in vector field: This is used to compute the work, flow, circultion, flux. Let r(t) x(t)î + y(t)ĵ + z(t)k; t b be spce curve, nd F (x, y, z) M(x, y, z)î + N(x, y, z)ĵ + P (x, y, z)k be vector field. () Work: The work done while moving prticle from the point r() to the point r(b) is given by F d r F (x(t), y(t), z(t)) d r dt dt There is list of equivlent formul given in the Tble (M dx + N dy + P dz). (4) (b) Flow: Flow of the vector field F long the curve r(t) from the point r() to the point r(b) is given by F d r. (5) Note tht mthemticlly (4) nd (5) re the sme. If the curve strts nd ends t the sme point i.e., r() r(b), then the flow is clled the circultion of the vector field long the curve. (c) Flux: Flux of vector field F (x, y) M(x, y) î + N(x, y) ĵ cross simple closed curve r(t) x(t) î + y(t) ĵ is given by (M dy N dx). (6) indictes tht the curve is closed curve. 4

5 onservtive Vector Field: Let f(x, y, z) be sclr function, then the grdient vector field of f is denoted by f nd defined s f f x î + f ĵ + f k. (7) f gives the orthogonl vector to the surfce defined by f(x, y, z) c for ny constnt c. A vector field F (x, y, z) M(x, y, z)î + N(x, y, z)ĵ + P (x, y, z)k is conservtive There exists potentil function f such tht F f. F. F d r, for ny simple closed curve. B A F d r is independent of the pth which connects the two points A nd B (8) [ indictes tht the sttements re equivlent, F is defined in (19)]. Finding the Potentil Function Let f(z, y, z) be the potentil function of the conservtive vector field F (x, y, z) M(x, y, z)î + N(x, y, z)ĵ + P (x, y, z)k. Then f M(x, y, z) f(x, y, z) M(x, y, z) dx + 1 (y, z) x f N(x, y, z) f(x, y, z) N(x, y, z) dy + 2 (x, z) f P (x, y, z) f(x, y, z) P (x, y, z) dz + 3 (x, y). To find the f(x, y, z), dd the bove three equtions but ignore the repeted expressions. For exmple, if M(x, y, z) dx e x cos y + xyz (9) N(x, y, z) dy xyz + e x cos y (1) P (x, y, z) dz xyz + z2 2, (11) then f(x, y, z) e x cos y + xyz + z (12) Note tht e x cos y ws present in the equtions (9) nd (1) but it ws written only once in (12). imilrly, xyz ws present in ll of (9), (1), nd (11), but it ws written only once in (12). Alterntively, you my use the method described in the book. 5

6 If prticle is moving from the point A to the point B in conservtive force field, then the work done is independent of the pth nd by the fundmentl theorem of line integrl, (4) nd (8) become Work where f is the potentil function of F. B A F d r f(b) f(a), Green s theorem on plne: Let F M(x, y) î + N(x, y) ĵ be vector field. We cn lwys use the formul (4) to find the work done or flow nd (6) to find the flux (in xy-plne) in ny vector field, regrdless of whether it is conservtive or not. Green s theorem (circultion): If M nd N re differentible functions of x, y, nd is simple closed curve oriented counter-clockwise, then (4) cn be written s ( N F d r (M dx + N dy) x M ) da, (13) where is the region enclosed by the simple closed curve. da cn be tken s dy dx, dx dy, or r dr dθ s per convenience. We know tht if the vector field F is conservtive, then work done or flow long closed curve is zero. Becuse if F is conservtive, then F. Which implies tht N x M. As result, from (13), we hve F d r. Green s theorem (flux): Let be simple closed curve in the xy-plne nd be the region enclosed by, then (6) cn be written s ( M (M dy N dx) x + N ) da, (14) where da cn be tken s dx dy, dy dx, or r dr dθ. Note tht the theorem is lso vlid in xz nd yz plnes. For exmple, if F M(x, z) î + N(x, z) ĵ nd is simple closed curve in xz-plne, then ( N (M dx + N dz) x M ) da, where is the region enclosed by the curve nd da dx dz dz dx r dr dθ. circultion density nd flux density: Let F M(x, y) î + N(x, y) ĵ be vector field. ircultion density t (x, y) N x M Flux density t (x, y) M x + N. Note tht (13) nd (14) re mthemticlly sme formul. For exmple, we cn derive (14) from (13) (M dy N dx) [( N) dx + M dy] [ ( M x N )] da (using (13)) ( M x + N ) da. 6

7 imilrly, (13) cn be derived from (14). Are of the region enclosed by simple closed curve: Let be the region enclosed by the simple closed curve. Then pplying the Green s theorem we hve 1 dy y dx) 2 (x 1 ( x 2 x + ) da da o the formul 1 2 Are of. (x dy y dx) cn be used to find the re of. urfce Integrls: Like line integrls, there re bsiclly two types of surfce integrls; surfce integrl of sclr function nd surfce integrl in vector field. urfce integrl of sclr function: This is used to find the re, mss, centroid, moment of inerti of three dimensionl surfce. The generl frmework is given below. Let be three dimensionl surfce. For exmple, : x 2 + y 2 + z 2 2, z is the surfce of the upper hemisphere of rdius. Then surfce integrl of function G(x, y, z) is given by I G(x, y, z) dσ, (15) where dσ is the infinitesiml surfce re of (like dx dy on xy-plne). If G(x, y, z) 1, then (15) gives the surfce re of. If G(x, y, z) δ(x, y, z) is the density of, then (15) gives the mss of the surfce. If G(x, y, z) (x 2 + y 2 )δ(x, y, z), then (15) gives the moment of inerti of with respect to the z-xis. ee the tble 16.3 for complete set of formul nd compre those with the formul listed on pge 2 of this review. finding dσ: There re two methods to find dσ. Projection method: In this method, we tke the projection/shdow of the surfce on to one of the coordinte plnes. While tking the shdow, we need to keep in mind tht the projection () must not be line/curve, nd (b) there should not be ny overlpping. For exmple, consider the surfce : x 2 + y 2 1, y, z 2. (16) This is hlf of cylindricl surfce which is stnding on the curve x 2 + y 2 1 nd 2 unit tll towrds the positive z-xis. We must not tke the shdow of this surfce on the xy-plne. Becuse then we will only get the curve x 2 + y 2 1 which violets the first requirement. We my tke the shdow on xz-plne (or yz-plne). But the shdows of x ± 1 y 2 will overlp on yz-plne. o, we should tke the shdow on the xz plne. The shdow on the xz-plne is the rectngle 1 x 1, z 2. Let f(x, y, z) be the eqution of the surfce, nd R be the shdow of the surfce on xy-plne. Then dσ is given by dσ f f da, k 7

8 where da cn be tken s dx dy, dy dx, or r dr dθ s per convenience. The limits of the integrtion re determined by the shdow R. For other coordinte plnes dσ dσ f f da, ĵ (xz-plne, da dx dz dz dx r dr dθ) f f da, î (yz-plne, da dy dz dz dy r dr dθ). In our exmple (16), f(x, y, z) x 2 + y 2 1. Therefore f 2x î + 2y ĵ. If we tke the shdow on the xz-plne then f ĵ 2y, nd the shdow is R : 1 x 1, z 2. Then the eqution (15) becomes I G(x, 1 x 2, z) 2 x 2 + y 2 da 2y R G(x, 1 x 2 1, z) dz dx. 1 x 2 Note tht y is replced by 1 x 2 which is obtined from the eqution of the surfce (16). Prmetriztion method: In this method, the eqution of surfce is given by two prmeters. For exmple, : r(u, v) cos v ĵ + sin v î + u k, v π, u 2, (17) is prmetriztion of the sme surfce (16). Then dσ is given by dσ r u r v. The limits of the integrtion is determined by the limits of the prmetriztion. In the bove exmple, r 1. Therefore the integrtion (15) will become u r v I π 2 G(cos v, sin v, u) du dv. Note tht x, y, z in G re replced by the prmetriztion (17) of. urfce integrl in vector field: This is used to find the flux of vector field cross surfce. Let be surfce nd F be vector field. Then the flux of F towrds the unit norml vector n to the surfce is given by Flux ( F n) dσ, (18) where dσ cn be found using the method described bove. Finding n: If the eqution of the surfce is given in the implicit form f(x, y, z), then the unit norml vector is given by n ± f f. 8

9 If the eqution of the surfce is given in the prmetric form r(u, v), then n is given by n ± r u r v r u r v where the sign ± is chosen from the description of the problem., tokes Theorem: Let F M(x, y, z) î + N(x, y, z) ĵ + P (x, y, z) k be vector field. url of the vector field F is denoted by F nd defined s F î ĵ k x M(x, y, z) N(x, y, z) P (x, y, z) ( P N ) ( M î + P ) ( N ĵ + x x M ) k. (19) tokes Theorem: Let be the (piecewise smooth) counter-clockwise oriented boundry of surfce. Note tht is simple closed curve. Flow of vector field F long which is given by (5) cn be written s F d r ( F ) n dσ, (2) where n is the unit norml vector to the surfce nd the direction of n is determined by the orienttion of using right-hnd rule. tokes theorem is three dimensionl nlogue of Green s theorem of flow. More precisely, (13) cn be obtined from (2). ivergence Theorem: Let F M(x, y, z) î + N(x, y, z) ĵ + P (x, y, z) k be vector field, be closed surfce nd be the region enclosed by. ivergence of F is denoted by F nd defined s o not get confused between (7), (21), nd (19). F M x + N + P. (21) ivergence Theorem: The outwrd flux of the vector field F cross the closed surfce which is defined by (18) cn be written s ( F n) dσ ( F ) dv. Note tht indictes tht the surfce is closed. 9