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Maths can be cool !!

Learn about cylindrical, spherical, and rectangular coordinate systems, transformations, unit vector transforms, and surface area calculations in mathematics.

garydsmith
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Maths can be cool !!

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  1. Maths can be cool !! Y = 1.5sqrt(-abs(abs(x)-1)*abs(3-abs(x))/((abs(x)-1)*(3-abs(x))))(1+abs(abs(x)-3)/(abs(x)-3))sqrt(1-(x/7)^2)+(4.5+0.75(abs(x-.5)+abs(x+.5))-2.75(abs(x-.75)+abs(x+.75)))(1+abs(1-abs(x))/(1-abs(x))),-3sqrt(1-(x/7)^2)sqrt(abs(abs(x)-4)/(abs(x)-4)),abs(x/2)-0.0913722(x^2)-3+sqrt(1-(abs(abs(x)-2)-1)^2),(2.71052+(1.5-.5abs(x))-1.35526sqrt(4-(abs(x)-1)^2))sqrt(abs(abs(x)-1)/(abs(x)-1)) http://www.howtogeek.com/106221/

  2. Maths can be cool !!

  3. Coordinate Systems Cylindrical Spherical Rectangular

  4. q f z R z x y r f Coordinate Systems Cylindrical Spherical Rectangular Suitable coordinate system determined by symmetry

  5. q f z R z x y r f ^ ^ ^ ^ ^ ^ ^ ^ ^ q f f R x z z y r Coordinate Systems Cylindrical Spherical Rectangular Suitable coordinate system determined by symmetry

  6. x = rcosf y = rsinf r = (x2+y2) ^ f = tan-1(y/x) z z f x ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ f =-xsinf + ycosf r = xcosf + ysinf y = rsinf + fcosf x = rcosf - fsinf y r f f y ^ Cylindrical Rectangular ^ r -x x Cartesian  Cylinder z fixed 2D problem (x,y)  (r,f) f ^ ^ z out of paper plane

  7. z = Rcosq ^ z r = Rsinq ^ R R = (z2+r2) q f r q q = tan-1(r/z) z R q ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ q =-zsinq + rcosq r = Rsinq + qcosq R = zcosq + rsinq z = Rcosq - qsinq r f Cylindrical Spherical Cylinder  Sphere f fixed 2D problem (r,z)  (R,q) r ^ z R ^ q ^ f into paper plane

  8. z = Rcosq x = rcosf r = Rsinq y = rsinf R = (z2+r2) r = (x2+y2) = (z2 + x2 + y2) q = tan-1(r/z) f = tan-1(y/x) = tan-1([x2+y2]/z) = Rsinqcosf = Rsinqsinf Rectangular  Sphere

  9. q f z R z x y r f Coordinate Transformations Rectangular Cylindrical Spherical x = Rsinqcosf y = Rsinqsinf z = Rcosq q = cos-1(z/R) R = √(x2 + y2 + z2) f = tan-1(y/x) • x = rsinf • y = rcosf • r = (x2 + y2 + z2) • = tan-1(y/x)

  10. q f z R z x ^ ^ y r ^ ^ f x = Rsinqcosf + qcosqcosf - f sinf ^ ^ ^ ^ ^ ^ ^ ^ ^ y = r sin f + f cosf x = r cos f - f sinf r = R sin q + q cosq ^ ^ ^ ^ y = Rsinqsin f + qcosqsinf + f sinf ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ z = Rcosq - qsinq q f f R x z z y r Unit Vector Transformations Rectangular Cylindrical Spherical Similarly for the other way around

  11. dl = h1dx1x1 + h2dx2x2 + h3dx3x3 dz dl = dr r+ rdf f + dz z 2p h ∫ ∫ 0 0 a 2p h ∫ ∫ ∫ 0 0 0 Length, Area,Volume CYLINDER Vary r Vary f df rdf dr dr Vary z dV = rdrdfdz Side Area = r df dz = 2prh Volume = rdr df dz = pa2h

  12. dl = h1dx1x1 + h2dx2x2 + h3dx3x3 dR R q dq Length, Area,Volume SPHERE Vary R Vary dq Rdq Vary df

  13. dl = h1dx1x1 + h2dx2x2 + h3dx3x3 dl = dR R+ Rdq q + Rsinqdff r = Rsinq q R p 2p ∫ ∫ 0 0 p a 2p ∫ ∫ ∫ 0 0 0 Length, Area,Volume SPHERE dV = dR.Rdq.Rsinqdf dR rdf Side Area = rdf.Rdq Rdq = R2 df sinqdq = 4pR2 Volume = dR.Rdq.Rsinqdf = R2dR. sinqdq. df = (4/3)pR3

  14. dl = dR R+ Rdq q + Rsinqdff ^ ^ ^ ^ ^ f r q f R dl = dr r+ rdf f + dz z Length, Area,Volume Where do these pre-factors come from? They arise because some unit vectors are themselves changing from point to point!! z y x

  15. ^ ^ ^ dr = dxx + dyy + dzz + xdx + ydy + zdz z ^ ^ ^ x y ^ ^ ^ x z y Length, Area,Volume ^ ^ ^ r = xx + yy + zz Rectangular

  16. ^ ^ r = rr + zz ^ ^ ^ r = xcosf + ysinf f = ycosf - xsinf ^ ^ ^ ^ dr = drr + rdr + dzz + zdz ^ ^ ^ ^ ^ f r ^ ^ ^ ^ ^ dl = dr r+ rdf f + dz z Length, Area,Volume z y dr = -xsinfdf + ycosfdf = (ycosf-xsinf)df = fdf x ^ The funny terms thus arise from the variation of the unit vector orientations !!

  17. q f z R z x y r f    Length, Area,Volume Spherical Rectangular Cylindrical dl2 = dR2 + R2df2 + R2sin2qdz2 dAr = (Rdq).(Rsinqdf) dAq= dR.(Rsinqdf) dAf = dR.Rdq dV = R2sinqdR.dq.df dl2 = dx2 + dy2 + dz2 dAz = dx.dy dAx= dy.dz dAy = dz.dx dV = dx.dy.dz dl2 = dr2 + r2df2 + dz2 dAz = dr.(rdf) dAr = dz.(rdf) dAf = dz.dr dV = rdr.df.dz 3 sets of surfaces

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