Example 2

Cylindrical Coordinates are defined by
x=rcos(theta)
y=rsin(theta)
z=z
r>=0, 0<=theta<=2Pi


  Clear[x,y,z,u,v,w]
  x[r_,t_,z_]:=r Cos[t]
  y[r_,t_,z_]:=r Sin[t]
  z[r_,t_,z_]:=z


  x[r,theta,z]
  y[r,theta,z]
  z[r,theta,z]


  Clear[a,b,c]
  {a,b,c}=Simplify[scalefactors[x,y,z][r,theta,z]]

Let's make a coordinate change on a vector field.


  Clear[F]
  F[x_,y_,z_]:={x/(x^2+y^2),y/(x^2+y^2),0}


  newF[r_,theta_,z_]:=
  F[x[r,theta,z],y[r,theta,z],z[r,theta,z]]


  Simplify[newF[r,theta,z]]

What are the divergence and curl?


  newdiv[newF][r,theta,z]

Yuck!


  Simplify[%]


  newcurl[newF][r,theta,z]


  Simplify[%]

Calculate the divergence and curl in rectangular coordinates.


  F[x_,y_,z_]:={x/(x^2+y^2),y/(x^2+y^2),0}


  curl3[F,x,y,z]


  div3[F,x,y,z]

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