Sharjah Institute of Technology

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1 For commets, correctios, etc Plese cotct Ahf Abbs: Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet ALGERA Lws of Idices:. m m + m m. ( ).. 4. m m 5. Defiitio of logrithm: Chge of bse: If the log logb Log logb Lws of logrithms:. log (A ) log A + log. A log log A - log. log A.log A m 0 m 6. Qudrtic Formul: If + b + c 0 The Newto-Rphso Itertive Method: b ± b 4c If is the pproimte vlue for the rel root of the equtio: f() 0, the closer pproimtio,, c be obtied from: f ( ) f '( ) Where f ' ( ) is the grdiet of the curve t. GRAPHS Stright lie grphs. Stright lie lw. verticl chge Grdiet horizotl chge m + c m is the grdiet. Lest squre method: c is where the lie cuts the is m + c m + c

2 For commets, correctios, etc Plese cotct Ahf Abbs: Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet SERIIES Arithmetic series. th term T + ( + ) d Sum of terms S + l + d ( ) ( ( ) where first term, l lst term, d differece betwee terms. Geometric series. ( r ) th term T r Sum of terms S r Sum to ifiit S r where first term, r rtio betwee terms. McCluri Series: v 4 f '(0) f ''(0) f '''(0) f ' (0) f ( ) f (0) !!! 4! Tlor Series: 4 v ( ) f '( ) ( ) f ''( ) ( ) f '''( ) ( ) f ' ( ) f ( ) f ( ) !!! 4! TRIIGONOMETRY Idetities: si t θ cos θ sec θ cosec θ cot θ θ cosθ siθ cos θ + si θ + t θ sec θ cot θ + cosec θ si (-θ ) -si θ cos (-θ ) +cos θ t (-θ ) -t θ Compoud gle dditio d subtrctio: si (A + ) si A cos + cos A si si (A - ) si A cos - cos A si cos (A + ) cos A cos - si A si cos (A - ) cos A cos + si A si if R si (kt + α) si kt + b cos kt the: R cos α, b R si α, R ( + b ) d α t - (b/) Double gles: Si A si A cos A Cos A cos A - si A cos A - - si A t A T A t A tθ

3 For commets, correctios, etc Plese cotct Ahf Abbs: Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet MATRIICES. 4 A mtri is rr of umbers: is mtri 7 0 Addig b e f e b f + c d + + g h c+ g d + h Subtrctig b e f e b f c d g h c g d h Multiplig b e f e bg f bh c d + + g h ce + dg cf + dh Notice tht whe multiplig mtrices, A d A re ot ecessril equl. Idetit mtri I 0 0 d A A I b d b Iverse of d order mtri A is A c d A c b Determit of d order mtri A is c d A Δ ( d ) ( bc) or A P Q R Determit of rd order mtri b c is d e f b c c b or Δ P Q + R e f d f d e ( ) ( ) ( ) ( ) ( ) ( ) Crmer s rule To solve simulteous equtios with ukows usig determits: + b+ cz+ d 0 or Δ P bf ec Q f dc + R e db + b+ cz+ d 0 + b+ cz+ d 0 z b c d c d b d b c Δ Δ Δ Δ b c d c d b d b c b c d c d b d b c z Δ Δ Δ Δ Δz z Δ

4 For commets, correctios, etc Plese cotct Ahf Abbs: Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet COMPLEX NUMERS Z ( + b) r (cos θ + si θ ) r θ r e θ Where - Modulus, r z ( + b ) Argumet, θ T - (b/) Additio: ( + b) + (c + d) ( + c) + (b + d) Subtrctio: ( + b) - (c + d) ( - c) + (b - d) (Additios d subtrctios cot be performed i polr form) Multiplictio: z z r r ( θ + θ ) Divisio: z r ( θ θ ) z r Coversio Polr r θ to rectgulr ± b r cosθ, b r siθ Rectgulr ± b to polr r θ b b r + b tθ θ t The comple cougte of ( + b) is ( b) + b ( + b)( c Divide c + d ( c + d)( c cougte of bottom) d) d) c d + bc bd c cd + cd d ( c + bd) + ( d + bc) c + d (multipl top d bottom b comple Powers ( r θ) r θ Roots r θ r De Moivre's Theorem: (r θ ) r θ r (cos θ + si θ ) θ 4

5 For commets, correctios, etc Plese cotct Ahf Abbs: Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet GEOMETRY Ares d Volumes. Trigle Are bse perpediculr height bsic bcsi A csi Prllelogrm Are bse perpediculr height Trpezium Are ( + b) h c Pthgors's Theorem: b + c Circle π d Are π r 4 A b Circumferece π r π d Sphere 4 Volume ( π r ) Surfce Are 4π r Clider Volume π rh Curved surfce re π rh Coe Volume ( π rh ) where h verticl height Curved surfce re π rl where l slt height Prism Volume bse re perpediculr height Prmid Volume (bse re) (perpediculr height) C Circulr Mesure. π rdis 60 0 To covert rdis to degrees multipl b 60 π,to covert degrees to rdis multipl b π 60 Legth of rc rθ where θ is i rdis Are of sector r θ No right-gled trigles b c si si cos si + cos si A si sic cos cos si sec + t b + c bccos A cos cos cosec + cot b + c ccos cos si si ( cos ) t c + b bcosc cos t ( + cos t si( A± ) si Acos ± cos Asi cos( A± ) cos Acos m si Asi t A± t t( A± ) m t A t 5

6 For commets, correctios, etc Plese cotct Ahf Abbs: Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet DIIFFERENTIIAL CALCULUS Stdrd Derivtives: d d or f() or f'() or f() or f'() - si cos e e cos - si l t sec Product Rule: whe uv where u & v re fuctios of, the: d du v + u dv Quotiet Rule: whe du dv v u u d where u & v re fuctios of, the: v v Chi Rule or 'fuctio of fuctio': if u is fuctio of, the: d d du du d f ( + h) f ( h) Numericl Differetitio - poit equtio: h d Mimum or miimum vlues: If f() the 0 for sttior poits To fid whether the sttior poits re m, mi or poits of iflectio: d Differetite gi to get. If the vlue is positive, the poit is miimum If the vlue is egtive, the poit is mimum If the vlue is zero t the poit, < 0 o oe side d > 0 o the other, the poit is poit of iflectio Prtil Differetitio:Rte of chge du dv If z f(u, v,.) d,, deote the rte of chge of u, v,.., the the rte of dt dt chge of z, dz dz dz du dz dv is give b: dt dt du dt dv dt Smll chgesif z f(u, v,.) d du, dv deote smll chges i u, v,.., the the dz dz correspodig chge i z, dz is give b: dz du + dv +... du dv 6

7 For commets, correctios, etc Plese cotct Ahf Abbs: Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet INTEGRAL CALCULUS Stdrd Itegrls + + c (ecept where -) cos si + c + e e + c si - cos + c l + c sec t + c Itegrtio b prts: dv du If u d v re both fuctios of, the: u. uv v. Mid-Ordite Rule: Are b legth of the mid-ordites Trpezium Rule: Are b ( + ) + ( ) b Are (0 + ) + 4( ) + ( ) Volumes of Solids of Revolutio f is rotted through 60 bout Simpso's Rule: [ ] Volume of solid obect geerted whe curve ( ) the -is is V π Volume of solid obect geerted whe curve f ( ) is rotted through 60 bout the -is is V π T STATIISTIICS Me: f. f Stdrd Devitio: σ f.( ) f 7

8 For commets, correctios, etc Plese cotct Ahf Abbs: Shrh Istitute of Techolog echicl Egieerig Yer Thermofluids sheet DIIFFERENTIIAL EQUATIIONS First Order Differetil Equtios:If f () f ( ) If f () d f () dq dt kq kt the Q Ae (where A d k re costts) d Itegrtig Fctor Method: If + P Q i. Rerrge the equtio ito the form bove (if ecessr) ii. Idetif P & Q iii. Fid the Itegrtig Fctor I e P iv. Substitute I ito:.i IQ v. Itegrte the RHS (b prts or relevt substitutio) to give the Geerl Solutio. vi. Substitute vlues to get the Prticulr Solutio Secod Order Differetil Equtios (Homogeous tpe): i. If d d + b + c 0 (where, b & c re costts) ii. Rewrite the equtio s (D + bd + c) 0 iii. Substitute m for D d solve the uilir qudrtic equtio: m +bm + c 0 iv. If the roots re: ) Rel d differet, s m α d m β, the the Geerl Solutio is: β Ae + e b) Rel d equl, s m α twice, the the Geerl solutio is: α ( A + ) e c) Comple, s m α ± β, the the Geerl solutio is: α e ( Acos β + si β) v. Fid the Prticulr solutio b substitutig vlues of, d d/ ito the Geerl solutio d it's derivtive. Secod Order Differetil Equtios (No-Homogeous tpe): i. If d d + b + c f () (where, b & c re costts) ii. Solve the complemetr fuctio c (see bove) iii. Use the followig tril fuctios for the prticulr itegrl p : If F ( ) k ssume C If F ( ) k C + D If F ( ) k ssume C + D + c If k F ( ) Ce k Ce If F ( ) C si k or D cos k C cos + Dsi 8