MATH 13 FINAL TUY GUI, WINTR 2012 This is ment to be quick reference guide for the topics you might wnt to know for the finl. It probbly isn t comprehensive, but should cover most of wht we studied in this clss. There re no exmples, so be sure to consult your old homeworks, notes, or textbook for those. 1. ontent from erlier clsses (vectors, differentition, etc.) Know bout bsic opertions on vectors, such s ddition, sclr multipliction, dot product, cross product, nd their pplictions to geometric problems Know bout properties of lines nd plnes in R 3 Know how to prmeterize curves, compute rc length, nd find tngent lines to curves Know bout prtil derivtives nd their pplictions to computing tngent plnes to surfces Know bout the grdient nd directionl derivtives 2. ouble integrtion Integrtion over rectngles Integrtion over more generl domins in the plne How to interchnge order of integrtion (Fubini s Theorem) Polr coordintes Applictions of double integrtion to finding re, mss of lmin, center of mss, etc. 3. Triple integrtion Integrtion over rectngulr prisms Integrtion over more generl three-dimensionl solids How to interchnge order of integrtion (Fubini s Theorem) ylindricl coordintes phericl coordintes Jcobin, chnge of vribles formul (this covers both double nd triple integrls) Applictions of triple integrtion to finding volume, mss of solid, center of mss, etc. 4. Line integrtion, vector fields Prmeterizing common curves (line segments, grphs of functions y = f(x), circles, ellipses, etc) lculting line integrls of sclr functions over curves lculting line integrls of vector fields over curves The Fundmentl Theorem of lculus for line integrls The vrious properties of conservtive vector fields 1
2 MATH 13 FINAL TUY GUI, WINTR 2012 How to pply the di erentil criterion P y = Q x for being conservtive vector field, simply connected domins Applictions of line integrls to clculting rc length, center of mss of wire, work, etc. Green s Theorem 5. urfce integrls Prmeterizing common surfces (plnes, cylinders, spheres, grphs of functions z = f(x, y), etc) lculting tngent plne nd norml vectors from prmeteriztion for surfce (in prticulr, using the fundmentl vector product) lculting surfce integrls of sclr functions over surfce Understnding wht orienttion on surfces mens lculting surfce integrls of vector fields over surfce lculting divergence nd curl of vector fields in R 3 How to pply the di erentil criterion r F = 0 for being conservtive vector field, simply connected domins The ivergence Theorem tokes Theorem Applictions of surfce integrls to finding surfce re, etc 6. Formuls A non-comprehensive collection of formuls from this clss: Polr coordintes: x = r cos,y = r sin, if is described by pple pple,g 1 ( ) pple r pple g 2 ( ), then f(x, y) da = g2 ( ) g 1 ( ) f(r cos,rsin )rdrd. ylindricl coordintes: x = r cos,y = r sin,z = z, if is described by pple pple,r 1 pple r pple r 2,z 1 pple z pple z 2 (z i my be function of r,, while r i might be function of ), then f(x, y, z) dv = r2 z2 r 1 z 1 f(r cos,rsin,z)rdzdrd. phericl coordintes: x = sin cos,y = sin sin,z = cos,if is described by 1 pple pple 2,..., then f(x, y, z) dv = 2 2 2 1 1 1 f( sin cos, sin sin, cos ) 2 sin d d d. Jcobin: If T is function from the uv plne to the xy plne, with x = x(u, v),y = y(u, v), then the Jcobin of T is defined to be
MATH 13 FINAL TUY GUI, WINTR 2012 3 J(u, v) = @(x, y) @(u, v) = x u x v y u y v = x u y v x v y u. The bsolute vlue of the Jcobin represents locl re mgnifiction fctor. There is corresponding formul for trnsformtion from uvw spce to xyz spce, involving 3 3determinnt. hnge of vribles formul: if is some region in the uv plne, nd R = T () (for exmple, if T is the polr chnge of coordintes, nd is the rectngle 0 pple r pple 1, 0 pple pple 2, then T () is the unit disc centered t the origin), then T () f(x, y) da = f(x(u, v),y(u, v)) J(u, v) du dv. There is corresponding formul for chnge of vribles from uvw spce to xyz spce. Line integrls of sclr functions: If is curve prmeterized by r(t) = hx(t),y(t)i,pple t pple b, ndf(x, y) somefunctionon, then f(x, y) ds = b f(r(t)) r 0 (t) dt = b f(x(t),y(t)) p x 0 (t) 2 + y 0 (t) 2 dt. When f(x, y) = 1,this specilizes to the formul for rc length of. There is corresponding formul for curves which lie in R n. Line integrls of vector fields: If is curve prmeterized s bove, nd F(x, y) somevectorfieldon, then F dr = b F(r(t)) r 0 (t) dt = b F(x(t),y(t)) hx 0 (t),y 0 (t)i dt. If F represents force, this line integrl equls the work this force does on prticle which moves long the pth. The Fundmentl Theorem of lculus for Line Integrls: if F = rf on curve prmeterized s bove, then F dr = f(r(b)) f(r()). onservtive vector fields: their properties. A vector field F is defined to be conservtive on domin if there exists function f on such tht F = rf. This is equivlent to the following properties: For ny closed curve in, F dr =0. For ny two curves 1, 2 in which strt nd end t the sme point, F dr = F dr. 1 2 If vector field FhP, Qi is conservtive, then P y = Q x in ll of. However, in generl the converse is true only if is simply connected. For F in R 3,the condition P y = Q x is replced by r F = 0.
4 MATH 13 FINAL TUY GUI, WINTR 2012 Green s Theorem: If is simple closed curve in the plne, its interior, F = hp, Qi some 1 vector field on, nd is given the positive (counterclockwise) orienttion, then F dr = P y da. Q x Fundmentl vector product: if surfce is prmeterized by r(u, v), the fundmentl vector product is the cross product r u r v, nd is in generl function of u, v. Its vlue t (u, v) isnormlvectortothesurfce t the point r(u, v), nd its length represents locl re mgnifiction fctor. urfce integrl of sclr function: if surfce is prmeterized by r(u, v) = hx(u, v), y(u, v), z(u, v)i, (u, v) 2, where is some domin in the uv plne, then f(x, y, z) d = f(r(u, v)) r u r v da = f(x(u, v),y(u, v),z(u, v)) r u r v da. When f(x, y, z) =1,thisspecilizestotheformulforthesurfcereof. urfce integrl of vector field: if is prmeterized s bove, given the orienttion which points in the sme direction s r u r v,ndf(x, y, z) is vector field on, then F d = F n d = F(r(u, v)) (r u r v ) da. The ivergence Theorem: if is some solid, the boundry of with outwrd orienttion, nd F 1 vector field over ll of, then F n d = r F dv. tokes Theorem: If is some oriented surfce, its boundry with induced orienttion from, ndf some 1 vector field on, then F dr = r F d. 7. trtegies for solving problems A list of strtegies we hve lerned for solving vrious types of problems. If you run into n iterted integrl which you cn t seem to evlute, try switching the order of integrtion or chnging coordinte systems. Polr coordintes re probbly useful when deling with circles, sectors of circles, nnuli, or other geometric figures relted to circles. ylindricl coordintes re good when deling with cylinders, prboloids, or cones. phericl coordintes re good when deling with spheres or pieces of spheres.
MATH 13 FINAL TUY GUI, WINTR 2012 5 When clculting the line integrl of vector field, you cn sometimes pply the Fundmentl Theorem of lculus for line integrls (if you cn find potentil function for F, which is not lwys possible or fesible) to skip prmeteriztion of the curve. You cn show tht F is conservtive on in vriety of wys. You cn either explicitly construct potentil function vi prtil integrtion, or check tht P y = Q x if is simply connected. You cn show tht F is not conservtive on in vriety of wys. You cn either check tht P y 6= Q x, or find closed curve in for which the integrl of F long is not equl to 0, or show tht no potentil function exists by trying to do prtil integrtion nd rriving t contrdiction. When clculting the line integrl of vector field over simple closed curve, sometimes Green s Theorem cn simplify the clcultion. This is especilly true if is polygonl, like rectngle, or some other curve which hs somewht complicted prmeteriztion. When clculting the surfce integrl of vector field, you cn sometimes skip lot of steps if you find out tht F n is constnt on nd is some geometric shpe which you cn esily clculte the re of. The ivergence Theorem is useful when you find surfce integrl of vector field which is complicted but whose divergence is simple, nd/or if the surfce encloses polyhedrl solid. tokes Theorem is sometimes useful for clculting line integrls if r F hs simpleform(likeconstntvectorfield),ndyoucnsometimesusetokes Theorem to interchnge the clcultion of surfce integrl F d of vector field F which equls r G for some G to the clcultion of F d, if, 0 hve the sme boundry curve. 0