Useful Formulae of Trigonometry

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1 Ueful Formule of Trigoometry DIFFERENTIATION, INTEGRATION AND FORMULAE USED CHAPTERWISE i q i q co q, co q co q, co q i q Agle i ( q) i q, co ( q) co q i 3 i (9 + q) co q (chge) co 3 i (p q) i q (No chge) t 3 3 HYPERBOLIC FUNCTIONS ih e e, coh d (ih ) coh, d (coh ) ih, i i i e e, co i e + e, coh ih i i e + e ih i i i, i ih i i, ch i co, coh co i Biomil Theorem ( + ) ( ) ( )( ) ! 3! Polr coordite r co q, y r i q, r + y, q y t r i q co f, y r i q i f, z r co q Medi i the lie joiig the verte to the mid poit of the oppoite ide of trigle. Cetroid or C.G. i the poit of iterectio of the medi of trigle. Icetre i the poit of iterectio of the biector of the gle of trigle. Circumcetre i the poit of iterectio of the perpediculr biector of the ide of trigle. Orthocetre i the poit of iterectio of the perpediculr drw from verte to the oppoite ide of trigle. Aymptote i the tget to curve t ifiity. (i)

2 DIFFERENTIAL CALCULUS d (co ) i d (t ) ec d (cot ) coec d d (ec ) ec t (coec ) coec cot d ( ) loge d d d (i ) (co ) (t ) + d d d (cot ) (ec ) (coec ) + d d d (ih ) coh (coh ) ih (th ) ec h d d d (coth ) coech (ech ) ech th (coech ) coech coth INTEGRAL CALCULUS log e i co co i t logec cot logi + log π ec log t + log(ec + t ) ih d coh 4 coec log t log(coec cot ) ec t ec coeccot coec i t + ih + cot + coec coh ih ec h th ech coech coth coech + i coh b b. f ( ) f () t dt. f ( ) ( ) 3. b f ( ) c f ( ) + b f ( ),( (, b)) c b coech coth coh ec ec h th ih f b b b f f b 4. ( ) ( + ) f ( ) f ( ) 6. f ( ) f () t f ( ) f ( ) + f ( ) if f ( ) (), if f ( ) f () (ii)

3 8. f ( ) f ( ), if f i eve fuctio i.e., f ( ) f (). f ( ) if f i odd fuctio i.e., f ( ) f (). CHAPTER (REVIEW OF VECTOR ALGEBRA) If r iˆ+ yj ˆ + zk ˆ the r + y + z AB mb + Poitio vector of B Poitio vector of A, Rtio formul c m+ Sclr Product: b. b coθ, Work doe F dr Vector Product b b i θη. ˆ. b. θ co b Are of prllelogrm b : Momet of force r F; V w r 3.( b c) b b b 3, c c3 ( b c) (. c) b( b. ) c c. d. ( b).( c d) bc. bd. Volume of prllelopiped.( b c), if.( b c) the bc,, re coplr. ( b) ( c d) [ bd] c [ bc] d [ bcd] + [ cd] b CHAPTER (DIFFERENTIATION OF VECTORS) d r Velocity dr dt, Accelertio dt Grdiet φ φ, Tget vector dr, Norml vector φ dt Directiol derivtive φ Divergece f. f, If divergece f, the f i clled oleoidl vector. Curl f f, If curl f, f i clled Irrottiol vector. CHAPTER 3 (INTEGRATION OF VECTORS) ψ φ Gree Theorem: φ +ψ dy dy c y, Stoke Theorem:. curl. ηˆ Gu theorem of Divergece: F. η d div f dy dz v F dr F d c CHAPTER 4 (ORTHOGONAL CURVILINEAR COORDINATES) Orthogol curvilier coordite. Let the rectgulr Crtei coordite of poit P i pce be (, h, z). Now we itroduce oe more ytem of coordite Let X (u, u, u 3 ), y Y (u, u, u 3 ), z Z (u, u, u 3 ) h r, h r, h r 3 u v v ds h du, dh h du, ds 3 h 3 du (iii)

4 CHAPTER 5 (DOUBLE INTEGRALS) Firt Method: b y f (, y) dy f (, y) dy A y Secod Method: d f (, y) dy f (, y) dy A c Chge of Order of Itegrtio O chgig the order of itegrtio, the limit of itegrtio chge. To fid ew limit, we drw the rough ketch of the regio of itegrtio. Some of the problem coected with double itegrl, which eem to be complicted, c be mde ey to hdle by chge i the order of itegrtio. CHAPTER 6 (APPLICATION OF THE DOUBLE INTEGRALS) Are, Volume d Surfce Z Are i Polr Co-ordite: Are r dθ dr S Z P γ N Surfce Are Let z f (, y) be the urfce S. Let it projectio o the -y ple be the regio A. Coider elemet. dy i the regio A. Erect cylider o the elemet. dy hvig it geertor prllel to OZ d meetig the urfce S i elemet of re. O A δ δy X Y. dy d co g, S A Triple Itegrtio It c be clculted CHAPTER 7 (TRIPLE INTEGRATION) z z + + dy y y z f(, y, z) dz dy, Firt we itegrte with repect to z tretig y z, y cott betwee the limit z d z. The reultig epreio (fuctio of, y) i itegrted with repect to y keepig cott betwee the limit y d y. At the ed we itegrte the reultig epreio (fuctio of oly) withi the limit d. Itegrtio by Chge of Crtei Coordite ito Sphericl Coordite Sometime it become ey to itegrte by chgig the crtei coordite ito phericl coordite. The reltio betwee the crtei d phericl polr co-ordite of poit re give by the reltio r i q co f y r i q i f z r co q dy dz J dr dq df r i q dr dq df (iv)

5 CHAPTER 8 (APPLICATION OF TRIPLE INTEGRATION) Volume of the olid Coider elemetry cuboid, whoe volume i dy dz. The the volume of the whole olid i obtied by evlutig the triple itegrl. If the equtio of the urfce of the olid be give i crtei coordite, the V dy dz (dv dy dz) dy rdqdr dydz r i q dr dq df Are Cetre of grvity y ' dy, Volume y y z dydz, dy y z (, y) drd θ (, r θ) ρ dy dz yρ dy dz zρ dy dz, y, z ρ dy dz ρ dy dz ρ dy dz Momet of Ierti bout i Momet of Ierti bout y i Momet of Ierti bout z i ρ + ( y z ) dy dz ρ + ( z ) dy dz ρ + ( y ) dy dz ρ d d y Cetre of preure, y ρ dy A yρ dy A ρ dy A CHAPTER 9 (GAMMA, BETA FUNCTION) +, +, π Gmm Fuctio: Bet Fuctio: b(l,m) Dirichlet Itegrl e Liouville Eteio of Dirichlet theorem m+ + ( ) m π l m, m i co θ θdθ l + m m+ + m lm y z dydz l + m+ + m lm h l + m+ f ( + y + z). y. z dydz f ( u) u du l + m+ + h Error fuctio ψα { φα } CHAPTER (THEORY OF ERRORS) l dt e π d ( ) d ψα ( ) f(, α) dφ dψ f (, y) f [ φ( α), α ] + f [ ψ( α), α] dα ( ) φ( α) α dα dα (v)

6 CHAPTER (FOURIER SERIES) f ( ) + co + co co bi + bi bi +... Where π π ( ), ( )co π f π f, π b ( )i π f π π For eve fuctio: ( ), ( )co, π f π f b π For odd fuctio:,, b ( ) i π f For rbitrry fuctio: c c ( ), ( ) co π f f c c c i p, co p ( ) c π b f ( )i c c CHAPTER (DIFFERENTIAL EQUATIONS OF FIRST ORDER) (i) Vrible eprble: f (y)dy f(), f ( y) dy φ ( ) + c (ii) Homogeeou Equtio: (iii) Reducible to homogeeou: dy f ( ) where ech term of f () d f() re of the me degree. φ ( ) dy dv Put y v o tht v+. dy + by + c A + By + C Put X + h, y Y + k if b A B, Put + by z if b A B Lier differetil equtio: dy + Py Q where P d Q re ot fuctio of y. Itegrtig fctor Ect Differetil Equtio If If M N y M N y p e P p, the y. e ( Qe ) + c the M + Ndy i ect differetil equtio. the the give differetil equtio i ot ect, but it c be reduced ect differetil equtio. M N y. If i fuctio of loe y f () the N. If If f ( ) e. O multiplyig the differetil equtio by itegrtig fctor become ect differetil equtio. N M y M i fuctio of y loe y f (y), the (vi)

7 I.F. f ( y) dy e o multiplyig the give differetil equtio by itegrtig fctor become ect differetil equtio. 3. If M i of the form M yf (y) d N i of the from N f (y) the I.F. 4. For thi type of m y (y + bdy) + m y ( y + b dy) the I.F. h y k. where m+ h+ + k + d b m' + h+ ' + k + ' b' O olvig thee equtio we get the Vlue of h d k.. M Ny CHAPTER 3 (LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER) Rule to fid Complemetry Fuctio m. whe root of A.E. m, m ; C.F. m ce + ce. whe root re equl; C.F. (c + c )e m 3. whe root re comple + ib ; C.F. e [c co b + c i b] Rule to Fid Prticulr Itegrl (i) (ii) (iii) e e, if f () ; e e if f () f( D) f( ) f( ) f ( ) [ f( D)], Epd [f (D)] d the operte f( D) i i co co f( D ) f( ) f( D ) f( ) If f ( ) the i i (iv) e. φ ( ) e. φ( ) f( D ) f ( ) f( D) f( D+ ) (v) φ ( ) f ( D) φ( ) f( D) f( D) f( D) (vi) φ ( ) e e φ( ) D+ CHAPTER 4 (CAUCHY EULER EQUATIONS, METHOD OF VARIATION OF PARAMETERS) Vritio of Prmeter C.F. Ay + By the P.I. uy + vy yx yx Where u v yy yy yy yy d y dy 4 Homogeeou Differetil Equtio i + 4y Put e z dy, Dy, d y d DD ( ) y, where D dz (vii)

8 CHAPTER 5 (LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER) Equtio of the type d y f ( ) Equtio of the type d y f ( y) Equtio which do ot Coti y Directly The equtio which do ot coti y directly, c be writte f d y d y dy,,...,...() dy d y dp O ubtitutig Pie.., d i (), we get P d f,... P, Equtio tht do ot Coti Directly The equtio tht do ot coti directly re of the form d y d y dy y f,,..., 3 d y d P 3 dy d y dp dp dy dp O ubtitutig P, P i the equtio (), we get dy dy dp,... P, y...() dy Equtio () i olved for P. Let dy dy P f( y) f( y) f( y) Equtio whoe oe Solutio i kow dy f ( y ) + c If y u i give olutio belogig to the complemetry fuctio of the differetil equtio. Let the other olutio be y v. The y u. v i complete olutio of the differetil equtio. d y dy Let + P + Qy R...(), be the differetil equtio d u i the olutio icluded i the complemetry fuctio of () d u du P Qu...() () y u. v dy du dv v + u d y v d u + dv du + u d v (viii)

9 dy d y Subtitutig the vlue of y,, i (), we get d u dv du d v du dv v + + u + P v u Qu. v R + + O rrgig d u du d v dv du dv v + P + Qu u P R The firt brcket i zero by virtue of reltio (), d the remig i divided by u. d v dv du dv P + + R u u d v du dv + P + u R u Norml Form (Removl of Firt Derivtive) d y dy Coider the differetil equtio + P + Qy R...() Put y uv where v i ot itegrl olutio of C.F. dy d y du du v + u u d v + du dv + v d u d dy d y O puttig the vlue of y,, i () we get d v dv du d u dv du u + + v P u v Q. uv R...(3) d u du dv d v dv v + Pv u P Q. v R d u du dv u d v dv + P P Qv v v R v Here i the lt brcket o L.H.S. i ot zero y v i ot prt of C.F. Here we hll remove the firt derivtive. dv dv P + or P or log v P v v P v e d v dv I () we hve to fid out the vlue of the lt brcket i.e., P Qv + + dv P P e Pv / P v e d v dp P dv dp P dp v v Pv v + P v 4 (i)...()

10 d v dv P Qv + + dp dp v + P v + P Pv + Qv v Q P 4 4 Equtio () i trformed d u u dp P + v Q v 4 Let d u dp P + u Q 4 Equtio (3) become R v Re P d u Qu + R where dp P Q Q 4 y uv d dv z, o tht d v dz P R Re or R v P v e A dz du R + P z + u u Thi i the lier differetil equtio of firt order d c be olved (z c be foud), which will coti oe cott. dv O itegrtio z, we c get v. Hvig foud v, the olutio i y uv. Note: Rule to fid out the itegrl belogig to the complemetry fuctio Rule Coditio u + P + Q e P + Q e 3 + P Q + e 4 P + Q 5 + P + Q 6 ( ) + P + Q CHAPTER 7 (APPLICATIONS TO DIFFERENTIAL EQUATIONS) Elemet Symbol Uit. Chrge q coulomb. Curret i mpere 3. Reitce, R + i ohm R ()

11 4. 5. Iductce, Cpcitce, + L i i hery H C + frd C 6. Electromotive force + E cott V or voltge (cott) i volt 7. Vrible voltge + E ~ vrible V volt i The formtio of differetil equtio for electric circuit deped upo the followig lw. dq (i) i, dt (ii) Voltge drop cro reitce R Ri (iii) Voltge drop cro iductce L. di L dt (iv) Voltge drop cro cpcitce C q C di Ri + L E dt R L L R di R + i E dt L L di R + i dt L q Ri c E i wt L i C + E E wt V C R i dq q R + dt c i Ri + dt E c i wt R E i wt di i R + E dt c co wt L C R d q dq + R + q dt dt c i i Mechicl Egieerig problem. V dt d, dt F m, dv V d F m, (i) dv F mv

12 Verticl motio Verticl Eltic Strig Horizotl Eltic Strig Stre (T) Stri Stri Stre Stri gt g ( kt e k k ) T Cott Module of Elticity (E) Etemio i legthq origil legth E l or E mg l Cott of elticity T E l d E Equtio of motio i m dt l Simple Hrmoic motio (S.H.M) d u dt The imple pedulum d Retorig force m dt mg i q d g i q dt Ocilltio of Sprig O O I A θ T P θ mg i θ mg O mg co θ d m dt mg k ( + ) Dmpled Free Ocilltio d + λ + u dt dt Forced ocilltio (without dmpig) d m dt mg k + q co t Forced ocilltio (without dmpig) Projectile y t d k k q + co t dt m m drt m g u co α A B P Y u i α k dt P mg u P (, y) A B e co t k ( + ) dmper O u co α A X (ii)

13 Fuctiol Euler equtio CHAPTER 8 (CALCULUS OF VARIATIONS) dy +. f d df y dy ' CHAPTER 9 (MAXIMA AND MINIMA OF FUNCTIONS) Workig rule to fid Etremum Vlue (i) Differetite f (, y) d fid out (ii) Put f d f f f f f,,,, y y y f y f (iii) Evlute r, f, t y y d olve thee equtio for d y. Let (, b) be the vlue of (, y). f for thee vlue (, b). (iv) If rt > d () r <, the f (, y) h mimum vlue. (b) r >, the f (, y) h miimum vlue. (v) If rt <, the f (, y) h o etremum vlue t the poit (, b). (iv) If rt, the the ce i doubtful d eed further ivetigtio. f f Note: The poit (, b), which re the root of d re clled ttiory poit. y Lgrge Method of Udetermied Multiplier Let f (, y, z) be fuctio of three vrible, y, z d the vrible be coected by the reltio. φ (, y, z)...() f (, y, z) to hve ttiory vlue, f, f y, f z f f f + dy + dz y z By totl differetitio of (), we get φ φ φ + dy + dz y z Multiplyig (3) by λ d ddig to (), we get...()...(3) f φ f φ f φ +λ + dy +λ dy + dz +λ dz y y z z (iii)

14 f φ f φ f φ +λ + +λ dy + +λ dz y y z z Thi equtio will hold good if f φ +λ f φ +λ y y f z φ +λ z...(4)...(5)...(6) O olvig (), (4), (5), (6), we c fid the vlue of, y, z d λ for which f (, y, z) h ttiory vlue. Drw Bck i Lgrge method i tht the ture of ttiory poit cot be determied. CHAPTER (COMPLEX VARIABLE FUNCTION) Alytic fuctio A igle vlued fuctio which i differetible t z z i id to be lytic t the poit z z u v u v C R Equtio:,. Ad u v ; u r v y y r r θ θ r v v To fid cojugte fuctio, dv + dy y Mile Thomo Method f (z) φ ( z,) dz + φ( z,) dz u u + dy if u i give y u u φ (, ), φ ( y, ) y, where y, where y f ( z) ψ ( z,) dz + ψ( z,) dz CHAPTER 3 (TRANSFORMATION) v v ψ (, ), ψ ( y, ). y For every poit (, y) i the z-ple, the reltio w f (z) defie correpodig (u, v) i the w-ple. We cll thi trformtio or mppig of z-ple ito w-ple z mp ito the poit w, w i lo kow the imge of z, If the poit P (, y) move log curve C i z-ple, the poit P (u, v) will correpodig curve C i w-ple, the we y tht curve C i the z-ple i correpodig curve C i the w-ple by the reltio w f (z). Coforml Trformtio Let two curve C, C, i the z-ple iterect t the poit P d the correpodig curve C, C i the w-ple iterect t P. If the gle of iterectio of the curve t P i z-ple i the me the gle of iterectio of the curve of w-ple t P i mgitude d ee, the the trformtio i clled coforml: coditio: (i) f (z) i lytic. (ii) f (z) If the ee of the rottio well the mgitude of the gle i preerved, the trformtio i id to be coforml. If oly the mgitude of the gle i preerved, trformtio i Iogol. Trltio: w z + c Rottio: w ze iq Or (iv)

15 Mgifictio: w c.z. Bilier Trformtio: w z + b cz + d Ivrit poit: w z + b the w z cz + d CHAPTER 4 (COMPLEX INTEGRATION) Cuchy Itegrl Theorem f ( z ) dz if f (z) i lytic fuctio withi C. c Cuchy Itegrl formul f ( z) dz ( ) c z πif, if f (z) i lytic i c, d i poit withi C. Reidue (i) Re f () lim ( z ) f ( z ), (ii) Re () φ( ) z ψ ( ) (iii) Re () d ( z ) f( z)!, (iv) Re () coefficiet of dz t Reidue Theorem f ( z ) dz πi (Sum of the reidue t the pole writte C) c π, where t z dz f(i θ, co θ) dθ, put i θ [ z ], co θ z+, dθ i z z iz C i the circle of rdiu oe. f( ), coider ( ) f( ) f z dz f( ) where f( ) c f( ) d c i the emicircle with rel i. CHAPTER 5 (TAYLOR S AND LAURENT S SERIES) Tylor erie. f (z) f () + f () (z ) + f "( )! (z ) f ( ) (z ) +...! Rdiu of Covergece : Luret theorem + lim R If we re required to epd f () bout poit where f (z) i ot lytic, the it i epded by Luret Serie d ot by Tylor Serie. Sttemet. If f (z) i lytic o c d c v d the ulr regio R bouded by the two cocetric circle c d c of rdii r d r (r < r ) d with cetre t, the for ll z i R f (z) + (z ) + (z ) b b z ( z ) f( w) πi c ( w ) + dw, r R A B W r b f ( w) πi c ( w ) + dw R C C (v)

16 Legedre Equtio CHAPTER 8 (LEGENDRE S FUNCTIONS) ( ) d y dy ( ) y () ( ) ( )( )( 3) 4 P ( ) () ()(3).4 Rodrigue formul Geertig Fuctio Orthogolity Property d P ( ) ( )! ( z + z ) ΣP ( ) z + P ( ). ( ) P m, if m d Recurrece Formule (i) ( + ) P+ ( + ) P P (ii) P P P (iii) (+ ) P P + P (iv) P P P (v) ( ) P [ P P ] (vi) Beel Equtio ( ) P ( + ) ( P+ P) CHAPTER 9 (LEGENDRE S FUNCTIONS) d y dy ( ) y + +, + P ( ) + r + r ( ) J( ) r r+ r+ Recurrece Formul (i) J J J + (ii) J J + J (iii) J J J + d (iv) J ( J + J+ ) (v) ( J ) J + (vi) d ( J ) J Hermite equtio CHAPTER 3 (HERMITE FUNCTION) d y dy + y Geertig fuctio of Hermite polyomil e { ( t ) } e t Orthogol property e Hm ( ) H ( ), H () + H + () t + H + () t +... m! π, m Recurrece Formule for H () of Hermite Equtio Four recurrece Reltio. H () H (). H () H () + H + () 3. H () H () H + () 4. H () H () + H () (vi)

17 CHAPTER 3 (LAGUERRES FUNCTIONS) d y dy + ( ) + y Lguerre Fuctio for Differet Vlue of. L ()! L () L () L () 4 + L 3 () L 4 () d o o. Geertig Fuctio of Lguerre Polyomil L ( ) ( t) t e! Recurrece Reltio t t t L ( ) t I. e ( t) t! II. L () L () + L () III. L + () + ( ) L () + L () t IV. ( t) e Orthogol Property L ( t )! / Let f( ) e L( )...()! L ( ) L ( ) m!! for m Over the itervl whe δ m, for m m f ( ) ( ) m f e δ m, CHAPTER 34 (LINEAR TRANSFORMATIONS) (i) T ( + b) T () + T (b), b U d (ii) T ( ) T () F, U Zero Trformtio if U (F) d V (F) be two vector pce over the me field F, the the mppig o ˆ,, U v. Mtri of Lier Trformtio Coider the imulteou equtio give below: (vii)

18 The left hd ide of the equtio c be coidered the lier trformtio of T T we c write the formul A:T A (X) AX y 5 C A (4, 5) S {(, ), (, )} S {(, ), (, )} 4 3 Pth (4, 5) 4 (, ) + 4 (, ) Pth (4, 5) 5 (, ) + 4 (, ) B 3 4 if T : A B i trformtio the et A i clled the domi d the et B i clled the codomi of T. Chge of Bi Codomi of lier trformtio Rk d Nullity of lier Trformtio Let T : V W be Lier Trformtio. We kow tht T (V) i ubpce of the vector pce. The dimeio of thi ubpce T (V) i clled the rk of T. Nullity of f dim [ker (f)] Similrity of Mtrice If A d B re qure mtrice of order over the field F, the B i id to be imilr to A, if there eit ivertible qure mtri P with elemet i F i uch tht B P AP CHAPTER 35 (BASIS OF NULL SPACE, ROW SPACE AND COLUMN SPACE) Row Vector Here we hve m mtri A... m m... m The elemet,... re kow eterie. If,... re rel, the eterie re i R. The row of A decribed vector i R re clled row vector or row mtri r (,, ) r (, ) r m ( m, m m ) (viii)

19 Colum Vector Colum vector of A re C..., C... m Colum pce m A {C, C... C } Row pce A [r, r... r } C m Null pce A i R i clled the ull pce of A. If i deoted by ull (A) Dimeio of Vector pce The umber of vector preet i bi of vector pce V i clled the dimeio of V. It i deoted by dim (V). Nullity The dimeio of the ull pce of the mtri A i clled the ullity of A d i deoted by ullity (A) or the umber of free vrible i the olutio of AX. Rk of Mtri The row rk of mtri i equl to the dimeio of the row pce of the mtri. The colum rk of mtri i equl to the dimeio of the colum pce of the give mtri. Rk Nullity Theorem Coider mtri A the rk (A) + ull (A) umber of colum of A. CHAPTER 36 (REAL INNER PRODUCT SPACES) Ier Product pce A vector pce together with ier product defied o it i clled ier product pce. We kow the clr product of vector d b i R we c defie ier product of two-colum vector d b (, b) T.b. A rel vector pce V i clled rel ier product pce if it h the followig propertie:. Symmetry. (X, Y) (Y, X). Additivity. (X + Y, Z) (X, Z) + (Y, Z) 3. Lierity. C(X, Y) (C X, Y) (X, CY) 4. Poitivity. (X, X) d (X, X) if d oly if X. The legth or orm of vector X i V i defied by X ( X, X) (i)

20 Orthogol Vector (Perpediculr Vector) Accordig to Cuchy Schwrz iequlity ( ) XY, X Y Sclr product of two vector ( ) XY, X Y coθ Where θ i the gle betwee two vector X d Y If co θ, the (X, Y) Thee two vector X d Y re kow orthogol vector. Grm-Schmidt orthogolitio-proce Y X ( Y, X) Y ( Y, Y) Uitry Trformtio Y X ( X, Y ) Y Y X Y ( X, Y ) ( X, Y ) _ Y Y Y X Y Y A lier trformtio, Y AX, where A i Uitry (i.e., A i uch A θ A A A θ I ), i clled uitry trformtio. Theorem. The ecery d ufficiet cditio for lier trformtio Y AX d V (C) to preerve legth i tht A i uitry. Orthogol Trformtio A trformtio Y AX i id to be orthogol if it mtri i orthogol. Orthogol projectio Projectio of log b b b b Lier Trformtio of Mtrice Let X d Y be two vector uch tht y y X, Y y y [,,..., ] y [ ] ()

21 Mior CHAPTER 37 (DETERMINANTS) The mior of elemet i defied determit obtied by deletig the row d colum cotiig the elemet. Thu the mior of, b d c re repectively. b c c b, d b3 c3 3 c3 3 b3 b c Thu b c (mior of ) b (mior of b ) + c (mior of c ). 3 b3 c3 Cofctor Cofctor ( ) r+c Mior Propertie of Determit Property (i). The vlue of determit remi ultered; if the row re iterchged ito colum (or the colum ito row). Property (ii). If two row (or two colum) of determit re iterchged, the ig of the vlue of the determit chge. Property (iii). If two row (or colum) of determit re ideticl, the vlue of the determit i zero. Property (iv). If the elemet of y row (or colum) of determit be ech multiplied by the me umber, the determit i multiplied by tht umber. Property (v). The vlue of the determit remi ultered if to the elemet of oe row (or colum) be dded y cott multiple of the correpodig elemet of y other row (or colum) repectively. Fctor Theorem If the elemet of determit re polyomil i vrible d if the ubtitutio mke two row (or colum) ideticl the ( ) i fctor of the determit. Whe two row re ideticl, the vlue of the determit i zero. The epio of determit beig polyomil i vihe o puttig, the i it fctor by the Remider theorem. CHAPTER 38 (ALGEBRA OF MATRICES) Type of equtio, AX B; C {A, B} () Coitet equtio If Rk of C Rk of A () Uique olutio If Rk of C Rk of A Number of ukow (b) Ifiite olutio If Rk of C Rk of A < Number of ukow () Icoitet equtio. If Rk of C Rk of A. Eige vlue re the root of the chrcteritic equtio A li O Cyley Hmilto Theorem. Every qure mtri tifie it ow chrcteritic equtio. Digolitio P AP D, where P i the modl mtri cotiig eige vector, D i the digol mtri cotiig eige vlue. (i)

22 Determit Crmer rule. olve the followig equtio. + b y + c z d + b y + c z d 3 + b 3 y + c 3 z d 3 D b c b c 3 3 c3, D d b c d b c, d3 b c3 D d c d c, 3 d3 c3 D 3 b d b d 3 b3 d3 D, D D y, D D3 z D Rk of Mtri The rk of mtri i id to be r if CHAPTER 39 (RANK OF MATRIX) () It h t let oe o-zero mior of order r. (b) Every mior of A of order higher th r i zero. Note: (i) No-zero row i tht row i which ll the elemet re ot zero. (ii) The rk of the product mtri AB of two mtrice A d B i le th the rk of either of the mtrice A d B. (iii) Correpodig to every mtri A of rk r, there eit o-igulr mtrice P d Q uch tht PAQ I r Norml form (Coicl form) By performig elemetry trformtio, y o-zero mtri A c be reduced to oe of the followig four form, clled the Norml form of A : (i) I r (ii) [I r ] (iii) I r The umber r o obtied i clled the rk of A d we write ρ(a) r. The form (iv) I r I r CHAPTER 4 (CONSISTENCY OF LINEAR SYSTEM OF EQUATIONS AND THEIR SOLUTION) Homogeeou Equtio For ytem of homogeeou lier equtio AX O (i) X O i lwy olutio. Thi olutio i which ech ukow h the vlue zero i clled the Null Solutio or the Trivil olutio. Thu homogeeou ytem i lwy coitet. A ytem of homogeeou lier equtio h either the trivil olutio or ifiite umber of olutio. (ii) If R (A) umber of ukow, the ytem h oly the trivil olutio. (iii) If R (A) < umber of ukow, the ytem h ifiite umber of o-trivil olutio. (ii)

23 A ytem of homogeeou lier equtio AX O Alwy h olutio Fid R(A) R (A) (o. of ukow) Uique or trivil olutio (ech ukow equl to zero) R (A) < (o. of ukow) Ifiite o. of o-trivil olutio Lier Depedece d Idepedece of Vector Vector (mtrice) X, X,... X re id to be depedet if () ll the vector (row or colum mtrice) re of the me order. () clr λ, λ,... λ (ot ll zero) eit uch tht λ X + λ X + λ 3 X λ X Otherwie they re lierly idepedet. Remember: If i et of vector, y vector of the et i the combitio of the remiig vector, the the vector re clled depedet vector. A ytem of o-homogeeou lier equtio AX B Fid R(A) d R(C) R (A) R(C) Solutio eit, ytem i coitet R (A) R (C) No olutio, ytem i icoitet R (A) R(C) (o. of ukow) Uique olutio R (A) R(C) < (o. of ukow) Ifiite o. of olutio Prtitioig of Mtrice Sub mtri. A mtri obtied by deletig ome of the row d colum of mtri A i id to be ub mtri. 4 For emple, A 5, the ,, re the ub mtrice. Prtitioig: A mtri my be ubdivided ito ub mtrice by drwig lie prllel to it row d colum. Thee ub mtrice my be coidered the elemet of the origil mtri. (iii)

24 : 4 : 3 4 For emple, A : : 6 5 CHAPTER 4 (EIGEN VALUES, EIGEN VECTOR, CAYLEY HAMILTON THEOREM, DIAGONALISATION) Fige Vlue Let X be uch vector which trform ito λx by me of the trformtio (). Suppoe the lier trformtio Y AX trform X ito clr multiple of itelf i.e. λx. AX Y λ X AX λ IX (A λi) X...() Thu the ukow clr λ i kow eige vlue of the mtri A d the correpodig o zero vector X eige vector. (b) Chrcteritic Polyomil: The determit A λi whe epded will give polyomil, which we cll chrcteritic polyomil of mtri A. For emple; λ 3λ λ ( λ) (6 5 λ + λ ) ( λ ) + ( 3 + λ) λ λ λ + 5 (c) Chrcteritic Equtio: The equtio A λi i clled the chrcteritic equtio of the mtri A e.g. λ 3 7λ + λ 5 (d) Chrcteritic Root or Eige Vlue: The root of chrcteritic equtio A λi re clled chrcteritic root of mtri A. e.g. λ 3 7 λ + λ 5 (λ ) (λ ) (λ 5) λ,, 5 Chrcteritic root re,, 5. Cyley-Hmilto Theorem Stemet. Every qure mtri tifie it ow chrcteritic equtio. If ( ) ( A I λ λ + λ + λ + + ) A ( ij ), the the mtri equtio Power of mtri i tified by X A i.e X X X I A A A I PD PP A PP A be the chrcteritic polyomil of mtri A PD P where D (iv) λ λ λ3

25 Workig procedure (i) Fid the eigee vlue of qure mtri A. (ii) Fid the correpodig eige vector d write modl mtri P. (iii) Fid digol mtri D from D P AP (iv) Obti A from A PD P Chrcteritic Vector or Eige Vector AX lx X i clled eige vector. Propertie of Eige Vector. The eige vector X of mtri A i ot uique.. If λ, λ,..., λ be ditict eige vlue of mtri the correpodig eige vector X, X,..., X form lierly idepedet et. 3. If two or more eige vlue re equl it my or my ot be poible to get lierly idepedet eige vector correpodig to the equl root. 4. Two eige vector X d X re clled orthogol vector if X X. 5. Eige vector of ymmetric mtri correpodig to differet eige vlue re orthogol. Orthogol Vector Two vector X d Y re id to be orthogol if T T X X X X. Algebric Multiplicity Algebric multiplicity of eige vlue i the umber of time of repetitio of eige vlue. 3 6 Geometric Multiplicity re 3, 3, 5. Geometric multiplicity of eige vlue i the umber of lierly idepedet eige vector correpodig to λ. It i deoted by Mult g (λ) 3 λ 3 re 3 d. So the mult g ( 3) Similrity Trformtio Let A d B be two qure mtrice of order. The B i id to be imilr to A if there eit oigulr mtri P uch tht B P AP...() Equtio () i clled imilr trformtio. Digolitio of Mtri Digolitio of mtri A i the proce of reductio of A to digol form D. If A i relted to D by imilrity trformtio uch tht D P AP the A i reduced to the digol mtri D through modl mtri P. D i lo clled pectrl mtri of A. (v)

26 Power of Mtri (By Digolitio) We c obti power of mtri by uig digolitio. We kow tht D P AP Where A i the qure mtri d P i o-igulr mtri. D (P AP) (P AP) P A (P P ) AP P A P Similrly D 3 P A 3 P I geerl D P A P...() Pre-multiply () by P d pot-multiply by P P D P P (P A P) P (P P ) A (P P ) A Procedure: () Fid eige vlue for qure mtri A. () Fid eige vector to get the modl mtri P. (3) Fid the digol mtri D, by the formul D P AP (4) Obti A by the formul A P D P. Hermiti Mtri Defiitio. A qure mtri A [ ij ] i id to be Hermiti if the (i, j)th elemet of A, i.e., For emple Skew-Hermiti Mtri ij 3+ 4 i b id, 3 4i b + id c ji for ll i d j. Defiitio. A qure mtri A ( ij ) i id to be Skew-Hermiti mtri if the (i, j)th elemet of A i equl to the egtive of the cojugte comple of the (j, i)th elemet of A, i.e., Periodic Mtri ij ji for ll i d j. A qure mtri i id to be periodic, if A k+ A, where k i poitive iteger. If k i the let poitive iteger for which A k+ A, the A i id to be of period k. Idempotet Mtri A qure mtri i id to be idempotet provided A A. Uitry Mtri A qure mtri A i id to be uitry mtri if θ θ AA AA I CHAPTER 4 (PARTIAL DIFFERENTIAL EQUATIONS) dy dz Pp + Qq R we c lo ue multiplier. P Q R z z z Homogeeou equtio + + A.E. i y m + m + y Ce I. If m m, m m, C.F. f (y + m ) + f (y + m ) Ce II. If m m C.F. f (y + m ) + f (y + m ) (vi)

27 (i) Prticulr Itegrl (ii) P.I. (iii) P.I. (iv) P.I. by by e + e +, f( DD, ) f( b, ) i( + by) i( + by) f ( D, DD, D ) f (, b, b ) (, ) f( DD, ) f y, ue Biomil Theorem f(, y) f(, c+ m) f ( D + md ) CHAPTER 43 (LINEAR AND NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS) No Homogeeou Equtio Moge Method Rr + S + Tt v dp dy dq r, t ubtitute the vlue of r d t i (ii). dy Rdy Sdy + T...(i) CHAPTER 44 (APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS) If (i) u u c t (wve equtio) (ii) u u c t (Oe dimeio het flow) (iii) u u + y (Two dimeio het flow) Clifictio z z z A + B + C + F (, y, z, p, q) y y. Prbolic if B 4A C. Elliptic if B 4A C < 3. Hypercritic if B 4A C > Fourier Trform CHAPTER 45 (INTEGRAL TRANSFORM) F () f () t f () t i dt. π i Fe () π Fourier Sie Trform f () F () i d F ( ) π t dt π f ( ) i (vii)

28 Fourier Coie Trform f () f () co F ( ) d π f ( t) co t dt. π. L(). 4. L (coh t) 7. L (co t). + CHAPTER 46 (LAPLACE TRANSFORM)! Lt ( ) L (ih t) t Le ( ) 6. L (i t) + 8. Le t f(t) F ( ) 9. Le f (t) Lf (t) f () L f ( t) Lf ( t) f () f (). d. Lt [ f( t)] ( ) [ F ( )] 3. d L f () t F ( ) d t 4. e 5. Lut [ ( )] 6. L[ f( t ). u( t )] 7. e F( ) T t 8. Lδ( t ) e e f () t dt 9. Lf () t T e. 3. Lt co t ( + ). L (i t t co t ) ( + ). L (i t t co t) 3 ( + ) t Covolutio Theorem L f () f( t ) F()* F() L f () t dt F( ) whe ut ( ) whe Lδ( t ) t < t > t L i t ( + ) CHAPTER 47 (INVERSE LAPLACE TRANSFORMS) L. L coh t 5. L co t 8. + t L 3. ( )! L ih t 6. t L F ( ) e ft () L 3 (i t t co t ). ( + ) L t co t. ( + ) L (i t + t co t ) ( + ) (viii) t L e L i t + L t i t ( + )

29 3. d L [ F( )] f ( t) + f () 4. dt t L F () f () t dt 5. L e F( ) f( t ) u( t ) 7. f() t L Fd () 9. t L f () f( t ) F(). F() t L F ( + ) e ft () d L F( ) tf() t d. f (t) um of the reidue of e t F () F( αi) αit F () t the pole of F (). L e G () i G ( αi) (i)