ENGI 2422 Appendix A Formulæ Page A-01

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1 ENGI 4 Appei A Formlæ Pge A- ENGI 4 Egieerig Mhemi Poiiliie for or Forml Shee Yo m ele iem from hi ome for pleme o or forml hee. However eigig or ow forml hee e vlle reviio eerie i ielf.. Fmel Eqio of ple hrogh poi P (where poiio veor of P) wih o-ero orml veor ABC : r i i or A B C D Eqio of lie hrogh poi P ( o o o ) (where poiio veor of P) prllel o o-ero veor v v v v3 : o o o r v or v v v3 If v he epre o he eqio o. If v he epre o he eqio o. If v 3 he epre o he eqio o. The i ge i priipl orml iorml veor poi o rve give r r() re r r T T T N B T N The r legh log he rve e fo from r The rvre κ i κ r r T T r N 3 r Coi Seio : ellipe mjor i mior i e < e < foi (±e ). ( i irle) : hperol verie (± ) mpoe ± / e > foi (±e ). 4 : prol vere ( ) fo ( ) e. i poi ( ); i he lie pir ± /.

2 ENGI 4 Appei A Formlæ Pge A- Qri Srfe : ellipoi (peil e re pheroi phere) : hperoloi of oe hee lige log i : hperoloi of wo hee lige log i : ellipi proloi : hperoli proloi : A igle poi he origi. : Nohig : Ellipi oe lige log he i; [mpoe o oh pe of hperoloi]. : Ellipi lier lige log he i. : Hperoli lier lige log he i. : Proli lier vere lie o he i. : Lie (he i) : Nohig : Ple pir (iereig log he i) : Prllel Ple Pir : Sigle Ple (he - oorie ple) : Nohig Srfe of Revolio f () roe ro. Eqio of rfe geere i ( ) ( ) ( ) f Are of rve rfe i ( ) ( ) ( ) A f f π.

3 ENGI 4 Appei A Formlæ Pge A-3 Trigoomeri ieiie Hperoli f ieiie θ e j oθ j iθ j j e e o oh j j j e e i j ih j i / o e / o / i o / o () o i () i () j e oh ih e e oh o j e e ih j i h ih / oh eh / oh h / ih oh / h oh () oh ih () ih h () h j o i e o o i ( i ) o ( ) ( ) e ( e ) e ( ) o ( o ) o (AB) o A o B i A i B o o i o i i (AB) i A o B o A i B i i o oh ih eh h h oh oh ih ( ih ) oh ( ) ( h ) eh ( eh ) eh h ( h ) h oh ( oh ) h oh (AB) oh A oh B ih A ih B oh oh ih oh ih ih (AB) ih A oh B oh A ih B ih ih oh

4 ENGI 4 Appei A Formlæ Pge A-4 Trigoomeri ieiie (o ) A B ( A B) A B o o i o i A o B (i(ab) i(ab)) / o A i B (i(ab) i(ab)) / o A o B (o(ab) o(ab)) / i A i B (o(ab) o(ab)) / i P i Q P Q P Q i o i P i Q P Q P Q o i o P oq P Q P Q o o o P oq P Q P Q i i Le ( / ) he i o Some Iegrl e C ( ) l C ( ) C > l ( ) ( ) e C f l f f ( ) ( ) C l e C o l i C e l e C l o C i ih C l C C h C l C C i C ih C l C

5 ENGI 4 Appei A Formlæ Pge A-5 Iegrio [or lr form] pr : v [ v ] v Some form h e oie from iegrio pr: ( l ) C l e i e ( i o) C e o e ( o i ) C m m m i o ( i o ( m ) i o ) m m m ( i o ( ) i o ) m A oher i-erivive h re reqire i qeio h o e oie from he ieiie ove will e pplie eiher irel or me of hi i he qeio. ( m ) Leii iff of iegrl: g( ) H( ) f ( ) g f H ( g( )) H ( f ( )) g( ) H( ) f ( )

6 ENGI 4 Appei A Formlæ Pge A-6. Pril Differeiio Chi rle: If f (... ) i g i (... m ) he j j j i j i i j i i i Grie: 3 ˆ ˆ ˆ I f f f f i j k Re of hge of f i he ireio of poi P i he ireiol erivive ˆ P P Df f i Joi (implii meho): Coverio from {... } o {... } efie impliil eqio f i ( ). Fi ll iffereil f i he or he mri eqio. e e The Joi i. A B B A Joi (eplii meho): ( ) ( ) ABS e Joi

7 ENGI 4 Appei A Formlæ Pge A-7 M-Mi: Chek ll poi: - o he omi or; - where f i efie; - where f i efie; - where f. Seo erivive e ( poi where f ): f f D f f D > f > lol miimm D > f < lol mimm D < le poi D : e fil. Lgrge Mliplier: Ieif fio f (... ) o e mimie or miimie. Ieif ori() g(... ) k. Solve he em of eqio f λ g g k. Solio wih mlle (lrge) vle of f i he miimm (mimm).

8 ENGI 4 Appei A Formlæ Pge A-8 3. Fir Orer ODE M( ) N( ) Seprle if M ( ) f ( ) g( ) N( ) ( ) v( ) Lier: P h h ( ) R( ) ; olio e ( e R C) where h P Berolli: [o i hi emeer] P( ) R( ) ; w ree o lier P ( ) ( ) w R( ) ig w E if M N ; olio ( ) where M N Iegrig For: Ue I() o r o mke ( ) Q( ) or P e: P Q ( ) l I (ivli if he iegr i epee o ). Q Ue I() o r o mke ( ) Q( ) P e: Q P ( ) l I (ivli if he iegr i epee o ). P Reio of orer (miig erm): To olve P ( ) R( ) Reple p reple Reio of orer (miig erm): To olve P Q R Reple p reple p p p

9 ENGI 4 Appei A Formlæ Pge A-9 4. Seo Orer Lier ODE P [P Q oh o]: Ailir eqio: Solve λ P λ Q λ λ λ Complemer fio: ( ) Q( ) R( ) Rel ii roo (over-mpe): Ae λ λ Be Rel repee roo (riill mpe): ( A B) e λ Comple ojge pir of roo (λ ± j) (er-mpe): j j e Ae Be ( ) ( o i ) e C D Prilr olio eermie oeffiie: If R() e k he r P e k If R() ( polomil of egree ) he r P ( polomil of egree ) wih ll ( ) oeffiie o e eermie. If R() ( mliple of o k /or i k) he r P o k i k B: if pr (or ll) of P i ile i he C.F. he mlipl P. Prilr olio vriio of prmeer: Le W A B he fi W R W R R R P W W v v W W v he Geerl olio: P Iiil (or or) oiio omplee olio. or e Lple rform.

10 ENGI 4 Appei A Formlæ Pge A- 5. Some Ivere Lple Trform F () f () F () f () e ( ) f ( ) ( ù) ( ù) f () π e e ( )! e δ ( ) e ω ( ) ω ( ) ( ) ( ) ω ( ) ( ) ( )! H ( ) i ω ω e i ω ω e ih e o ω e oh o ( ω ) ( ω ) { F () f () f N() 3 f O()... () f () f () () } ω ω ω i ω ω 3 ( ω ) ( ω ) ω ( ω ) h h F( ) F ( e ) i ω ω o ω 3 ω i ω ω o ω Sqre wve perio mplie Triglr wve perio mplie Swooh wve perio mplie f f ( τ ) τ f ()

11 ENGI 4 Appei A Formlæ Pge A- ( { }) Fir hif heorem: wih F( ) L f ( ) The ivere Lple rform of F ( ) i e f (). Seo hif heorem: The ivere Lple rform of e F () i f () H (). Slig proper ( eeio of he fir hif heorem): The ivere Lple rform of F ( ) i e f. Perio i fio : If f () i perioi fio of fmel perio T he T he Lple r form of f () i e f ( ). T e Sifig proper of he Dir el fio: f ( ) δ ( ) f ( ) < or > Covolio: If he Lple rform of fio f () g () re F () G () repeivel he he ivere Lple rform of H () F () G () i he ovolio h () (f * g)() (g * f )() f ( τ ) g( τ ) τ f ( τ ) g( τ ) τ Alo: F ( ) σ ( e ) σ F () f ( ) f ( )

12 ENGI 4 Appei A Formlæ Pge A- 6. Mliple Iegrio If he rfe ei i σ f ( ) he he m i h ( ) q ( ) m f ( ) f ( ) g( ) p( ) where he ier iegrl m e evle fir. Polr oorie: ( ) (r o θ r i θ) A r r θ. Cere of m i ( ) where m M m M. m σ A M σ A M σ A D D D Cliril polr oorie: ( ) (r o φ r i φ ) V r r φ. Spheril polr oorie: ( ) (r i θ o φ r i θ i φ r o θ) V r iθ r θ φ. M m V. ρ V Aiiol Formle for Polr Coorie (if eee) ( ) (r o θ r i θ ) r iθ θ θ r oθ θ θ Ar legh r θ r oθ r iθ r L r θ θ α β β Are wep o r f (θ ): A r θ α ˆ ˆ rˆ θ θ θ θrˆ v r rˆ r θ ˆ θ ( ) ˆ ( ) ˆ ( ) ˆ r r θ r r θ r θ θ r r θ r ( r θ) ˆ θ r END OF APPENDIX A