Applications of Vector analysis & PDEs

7/25/2019 Applications of Vector analysis & PDEs

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GROUP MEMBERS

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From2014-ME-01

To

2014-ME-20


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TOPIC

Practical application ofthe vector analysis


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WHAT WE HAE COERE!

Air Craft ectorin"

Solar panel

Use of c#rl

Use of "ra$ient vector

Cannon

Win$ vector

Sports %orce tore etc'((

Roller coaster


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Installation of Solar Panels

Solar panels have to 0e installe$ caref#lly so that

the tilt of the roof- an$ the $irection to the s#n-

pro$#ce the lar"est possi0le electrical po/er inthe solar panels(

A si.ple application of vector $ot an$ cross

pro$#cts lets #s pre$ict the a.o#nt of electrical

po/er the panels can pro$#ce(

A s#rveyor on the si$e/al3 #ses his instr#.entsto $eter.ine the coor$inates of the fo#r cornersof a roof /here solar panels are to 0e .o#nte$(


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Use of CUR4 for H#.an Welfare

V

V

Ass#.e /e insert s.all pa$$le /heels in a ,o/in"river(The ,o/ is hi"her close to the center an$ slo/er at

the e$"es(Therefore- a /heel close to the center 5of a river6/ill not rotate since velocity of /ater is thesa.e on 0oth si$es of the /heel(Wheels close to the e$"es /ill rotate $#e to$i7erence in velocities(

The c#rl operation $eter.ines the $irection an$ the.a nit#$e of rotation(


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Use of Gra$ient vector

A "ra$ient vector is the "ra$ient of a f#nction- /hich represents the$irectional chan"e in a scalar f#nction( It is cr#cial to $eter.ine if a vector*el$ is conservative(

The easiest /ay to $eter.ine /hether a vector *el$ is conservativeis 0y

$e.onstratin" that all the in*nite close$ c#rves in a vector *le$have no circ#lation /hich .eans the res#lt of their inte"ration is 8ero(

Another /ay of provin" a vector *el$ is conservative is 0y *n$in" the c#rl of the vector9 if the c#rl is a non:8ero val#e- this.eans the vector *el$ is path:$epen$ent (


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Cannon

A cannon is any piece of artillery that#ses "#npo/$er or other #s#allye;plosive:0ase$ propellants to la#nch a

pro


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Wind Vectors

4ets say /e have plane /ith constant velocity-an$ plane .ove to so#than$ /e have /in$ force/hich $irection of it is /est- so $#e to plane.ove.ent is so#th an$ /in$ .ove.ent is /est-

*nally plane .ove $ia"onally- or in the so#th:/est(E=AMP4E

S#ppose yo# are ri$in" a 0icycle on a $ay /henthere is no /in$( Altho#"h the /in$ spee$ is 8ero

yo# /ill feel a 0ree8e on the 0icycle $#e to the factthat yo# are .ovin" thro#"h the air( This is theapparent /in$( On the /in$less $ay- the apparent/in$ /ill al/ays 0e $irectly in front an$ eal inspee$ to the spee$ of the 0icycle(


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Sports (Baseball)

Another e;a.ple of a vector in real life/o#l$ 0e an o#t*el$er in a 0ase0all"a.e .ovin" a certain $irection for aspeci*c $istance to reach a hi"h ,y 0all0efore it to#ches the "ro#n$( Theo#t*el$er can2t


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Apparent wind in sailing

In sailin"- the apparent /in$ is the act#al ,o/ of airactin" #pon a sail( It is the /in$ as it appears to thesailor on a .ovin" vessel( It $i7ers in spee$ an$$irection fro. the tr#e /in$ that is e;perience$ 0y astationary o0server( In na#tical ter.inolo"y- these

properties of the apparent /in$ are nor.allye;presse$ in 3nots an$ $e"rees( On 0oats- apparent/in$ is .eas#re$ or >felt on face ? s3in> if on a $in"hyor loo3in" at any telltales or /in$ in$icators on:0oar$(

Tr#e /in$ nee$s to 0e calc#late$ or stop the 0oat(

Win$s#rfers an$ certain types of 0oats are a0le to sailfaster than the tr#e /in$( These incl#$efast .#ltih#lls an$ so.e plannin" .onoh#lls( Ice:sailors an$ lan$:sailors also #s#ally fall into thiscate"ory- 0eca#se of their relatively lo/ a.o#nt

of $ra" or friction


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7. Force, Torque, Acceleration,

Velocit and etc

%or calc#latin" every vectorial #nit- yo# nee$vector(

E;a.ple

There is a tire /ith .ass .- an$ it has initial an$

*nal velocity- acceleration- "ravitational-reaction- friction forces- an$ $#e to rotation it hastore( %or "ettin" the res#lt- yo# nee$ vectors(

In constr#ction- every architect have to 3no/

their 0#il$in"s of $#ra0ility- for this they nee$forces that .a; ho/ .any force /ill apply totheir 0#il$in"- an$ of co#rse they nee$ a"ainvectors( So yo# can see ho/ the vectors arei.portant


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!oller Coaster

A roller coaster is an a.#se.ent ri$e $evelope$for a.#se.ent par3s an$ .o$ern the.e par3s( Most ofthe .otion in a roller:coaster ri$e is a response to theEarth2s "ravitational p#ll(

@o en"ines are .o#nte$ on the cars( After the train

reaches the top of the *rst slope the hi"hest point on theri$e the train rolls $o/nhill an$ "ains spee$ #n$er theEarth2s "ravitational p#ll( The spee$ is s#+cient for it tocli.0 over the ne;t hill( This process occ#rs over an$ overa"ain #ntil all the train2s ener"y has 0een lost to friction

an$ the train of cars slo/s to a stop( If no ener"y /erelost to friction- the train /o#l$ 0e a0le to 3eep r#nnin" aslon" as no point on the trac3 /as hi"her than the *rstpea3(

Here vectors of forces- acceleration- an$ velocity are

i.portant to .a3e a safety syste.- if $esi"ner consi$erthe. acc#rately then syste. /ill 0e safety(


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Air "lane

If a plane is tryin" to ,y to an airstrip .iles north of itsc#rrent location- an$ it is travellin" at .ph- a person.i"ht ass#.e the plane /o#l$ arrive in D .in#tes if thepilot pointe$ the plane $#e north( B#t if the plane ha$ aprevailin" /in$ fro. the /est 0lo/in" at .iles per ho#r-

it /o#l$ en$ #p F .iles to the east of the airport- as the/in$ has 0lo/n the airplane o7 co#rse(

Thin3 of the plane2s spee$ an$ $irection as vector A- an$the /in$2s spee$ an$ $irection as vector B( A$$ the. hea$to foot- then $ra/ a line fro. the plane2s startin" point tothe en$ of vector B( This is the act#al location of the plane(

Win$s 0lo/in" at the plane fro. the front are calle$hea$/in$s- /hile /in$s fro. the 0ac3 of the plane arecalle$ tail/in$s( All /in$s fro. any $irection are calc#late$as vectors- an$ these vectors a7ect aircraft co#rses an$reire consistent co#rse correction(


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TOPIC

Practical applicationof the partial

$i7erential e&(


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Mathe.atical an$ Physical

Eations

P!Es are 3no/n to 0e /i$ely #se$ inthe $erivation of .any .athe.atical

an$?or physical eations or proofsan$ their i.portance can $e*nitelynot 0e ne"lecte$(


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Other Applications

P!Es are #se$ to *n$ o#t ho/ .#cha 0ea. is "oin" to 0en$(

P!Es are #se$ to *n$ the stress$istri0#tion /ithin an o0


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practical applications

%or practical applications- partial $i7erential eations"et translate$ into speci*c $i7erential eations(

More generally, computer software based onordinary dierential e!uations is used, rather

than solving the e!uations by hand every time. "eat e!uation in one space dimension

#ave e!uation in one spatial dimension

$enerali%ed heat&li'e e!uation in one space

dimension (pherical waves

)aplace e!uation in two dimension(


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Mar"inal Cost

*he cost added by producing one e+tra item ofa product is called marginal cost.

Many econo.ists #se the $i7erential eations to

calc#late the .ar"inal cost(

S#ppose it is state$ that the .ar"inal cost to.an#fact#re ; /i$"ets is "iven 0y the f#nction M C! ; C F( We can e;press this state.ent .ore

concisely /ith the $i7erential eation $C $; ! ; CF

/here C(;? $enotes the total cost to pro$#ce ; /i$"ets


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Applications of Vector analysis & PDEs

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