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MathGroup Archive 2003

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Re: Derivatives in Other Coord Systems

  • To: mathgroup at smc.vnet.net
  • Subject: [mg40249] Re: Derivatives in Other Coord Systems
  • From: sodastereo at eudoramail.com (Julius Carver)
  • Date: Fri, 28 Mar 2003 04:28:43 -0500 (EST)
  • References: <b5rm2k$etq$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

It's very easy Ken. First you have to define the position vector in a
coordinate system. For example in cylindrical coordinates

In[1]:=
<< Calculus`VectorAnalysis`

In[2]:=
Rc[t] = CoordinatesToCartesian[{r[t], q[t], z}, Cylindrical]

Out[2]=
{Cos[q[t]] r[t], r[t] Sin[q[t]], z}

or in spherical ones

In[3]:=
Rs[t] = CoordinatesToCartesian[{r[t], q[t], f[t]}, Spherical]

Out[3]=
{Cos[f[t]] r[t] Sin[q[t]], r[t] Sin[f[t]] Sin[q[t]], Cos[q[t]] r[t]}


Then you have to derive these expressions one or two times to obtain
the velocity or the acceleration respectively

In[4]:=
D[Rc[t], {t, 1}]

Out[4]=
{-(r[t] Sin[q[t]] q'[t]) + Cos[q[t]] r'[t], Cos[q[t]] r[t] q'[t] +
Sin[q[t]] r'[t], 0}

In[5]:=
D[Rs[t], {t, 2}]

Out[5]=
{2 Cos[q[t]] q'[t] (-(r[t] Sin[f[t]] f'[t]) + Cos[f[t]] r'[t]) + 
 
                                    2
   Cos[f[t]] r[t] (-(Sin[q[t]] q'[t] ) + Cos[q[t]] q''[t]) + 
 
                                                                2
   Sin[q[t]] (-2 Sin[f[t]] f'[t] r'[t] + r[t] (-(Cos[f[t]] f'[t] ) -
Sin[f[t]] f''[t]) +
 
      Cos[f[t]] r''[t]), 2 Cos[q[t]] q'[t] (Cos[f[t]] r[t] f'[t] +
Sin[f[t]] r'[t]) +
 
                                    2
   r[t] Sin[f[t]] (-(Sin[q[t]] q'[t] ) + Cos[q[t]] q''[t]) + 
 
                                                               2
   Sin[q[t]] (2 Cos[f[t]] f'[t] r'[t] + r[t] (-(Sin[f[t]] f'[t] ) +
Cos[f[t]] f''[t]) +
 
      Sin[f[t]] r''[t]), -2 Sin[q[t]] q'[t] r'[t] + 
 
                          2
   r[t] (-(Cos[q[t]] q'[t] ) - Sin[q[t]] q''[t]) + Cos[q[t]] r''[t]}


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