Polar Coordinates
Polar Coordinates. In a rectangular coordinate system we were plotting points based on an ordered pair of (x
4. Calculating ds in a different coordinate system Cylindrical polar
(c) Starting from ds2 = dx2 + dy2 + dz2 show that ds2 = d?2 + ?2d?2 + dz2. (d) Having warmed up with that calculation repeat with spherical polar coordinates
1 Return to List of Lessons Calculator Lesson 22 Polar Coordinates
Polar Coordinates and Graphs. The calculator can work in any one of three coordinate systems rectangular
4 2D Elastostatic Problems in Polar Coordinates
Feb 4 2018 Stresses and Strains in Cylindrical Coordinates ... A similar calculation can be carried out for forces in the tangential direction {? ...
Express the vector field A = yz i ? y j + xz2 k in cylindrical polar
Nov 10 2018 coordinates
THE CALCULUS OF CLOVERS
used multiple integration involving double and triple integrals in polar and cylindrical coordinates to calculate the areas and volumes of these shapes.
PHYS 631: General Relativity
May 16 2019 (Schutz 6.29) In polar coordinates
Cylindrical Coordinates
The unit vectors in the cylindrical coordinate system are functions of position. It is convenient to express them in terms of the cylindrical coordinates
April 8: Triple Integrals via Spherical and Cylindrical Coordinates
Apr 8 2020 We want a. 3-dimensional analogue of integrating over a circle. So we integrate over B
Integrals in cylindrical spherical coordinates (Sect. 15.7) Cylindrical
Remark: Cylindrical coordinates are just polar coordinates on the The calculation is simple the region is a simple section of a sphere.
4.Calculatingdsinadi↵erentcoordinates ystem
Cylindricalpolarcoordinatesaredefine dby
x=⇢cos y=⇢sin z=z (a)Confirmthatdx=d⇢cos⇢sind. (b)Calculateasimilarexpress ion fordy. (c)Startingfromds 2 =dx 2 +dy 2 +dz 2 showthatds 2 =d⇢ 2 2 d 2 +dz 2 (d)Havingwarmedupwit hthatcalculation,r epeatwi thsphericalp olarcoordinat eswhichare definedby x=rsin✓cos y=rsin✓sin z=rcos✓ andshow thatds 2 =dr 2 +r 2 d 2 +r 2 sin 2 d 2 Hint:Thespheric alresultis easiertogetstartingfromthe cylindricalresul tandus ing ⇢=rsin✓.4.Solution:Calculatingdsinadi↵erentcoordinatesy stem
(a)Thisisasimpl eapplic ati onoftheproductruledx=d⇢cos⇢sind. (b)dy=d⇢sin+⇢cosd. (c)Now dx 2 +dy 2 =(d⇢cos⇢sind) 2 +(d⇢sin+⇢cosd) 2Thecrosst ermscancelso
dx 2 +dy 2 =d⇢ 2 (cos 2 +sin 2 2 d 2 (sin 2 +cos 2 )=d⇢ 2 2 d 2Addingdz
2 givesthedesi redresult . (d)Startwithds 2 =d⇢ 2 2 d 2 +dz 2 .Now ⇢ 2 =r 2 sin 2 ✓andthe productru leappliedtod⇢ gives d ⇢=drsin✓+rcos✓d✓ andalso dz=drcos✓rsin✓d✓Thussubstit utingintods
2 fromabovegi ves ds 2 =(drsin✓+rcos✓d✓) 2 +r 2 sin 2 d 2 +(drcos✓rsin✓d✓) 2Onceagainthecr oss-ter mscancelle aving
ds 2 =dr 2 (sin 2 ✓+cos 2 ✓)+r 2 (cos 2 ✓+sin 2 ✓)d✓ 2 +r 2 sin 2 d 2 whichsimplifiesto thedesiredresultds 2 =dr 2 +r 2 d 2 +r 2 sin 2 d 2 45.GeodesicsontheSphere
Theequati onofasphereinspheri calp olarc oordinatesisparticul arlysim ple:itisr=a,where aisacons tant . (a)Startingwithdsinsphe ricalpolarcoordinates,writed ownthesim plifiedformofdswhen r=aisacons tant . (b)Usethis expressionfordstowri tedownanintegralth atrepresents the distancebetween twopoints connectedbyapatht hatliesonthesurfaceofaspher e.Wri tet heintegrali n theformwhe reisafun cti onof✓. (c)Writedownafirst integralforthi sin tegrand. (d)Showthat 0 =sin 1 [↵cot✓] satisfiesthefirstintegral ,where 0 and↵aretwo independe ntconstants. (e)Theequati onofaplanethroughtheorigi ni sAx+By+Cz=0.R ewr itethisequationin sphericalpolarcoordinates.Re arrangetheeq uationtomakeitlooklikethesolutionabo ve andfind ↵and 0 interm sofA,BandC. (f)Thusgiveasim plegeometri cde scriptionandmethodof findinggeodesicsonasphe re.5.Solution:GeodesicsontheS phere
(a)Ifr=aisacons tant then ds 2 =a 2 d 2 +a 2 sin 2 d 2 (b)Theintegr alis I= Z ds=a Z B A q 1+sin 2 0 2 dThusF(✓,,
0 p 1+sin 2 0 2 (c)Since@F/@=0 afir sti ntegralis sinquotesdbs_dbs12.pdfusesText_18[PDF] cylindrical coordinates examples
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