Example 1

Consider the curvilinear coordinate system x=u-v, y=u+v, w=z^2 for z>=0 (we need this to have an inverse!).

In[14]:=

  Clear[x,y,z,u,v,w]

In[15]:=

  x[u_,v_,w_]:=u-v
  y[u_,v_,w_]:=u+v
  z[u_,v_,w_]:=w^2

In[16]:=

  scalefactors[x,y,z][u,v,w]

Out[16]=

                             2
  {Sqrt[2], Sqrt[2], 2 Sqrt[w ]}

Answer:

Answer: ...

In[17]:=

  {a,b,c}=scalefactors[x,y,z][u,v,w]

Out[17]=

                             2
  {Sqrt[2], Sqrt[2], 2 Sqrt[w ]}

In[18]:=

  Clear[F]
  F[u_,v_,w_]:={u w,v w, u v}

In[19]:=

  newdiv[F][u,v,w]

Out[19]=

  Sqrt[2] w

In[20]:=

  newcurl[F][u,v,w]

Out[20]=

                            2                         2
   -(Sqrt[2] v) + 2 u Sqrt[w ]  Sqrt[2] u - 2 v Sqrt[w ]
  {---------------------------, ------------------------, 0}
                       2                           2
       2 Sqrt[2] Sqrt[w ]          2 Sqrt[2] Sqrt[w ]

In[21]:=

  Simplify[%]

Out[21]=

                     2                        2
      u      v Sqrt[w ]       v       u Sqrt[w ]
  {------- - ----------, -(-------) + ----------, 0}
   Sqrt[2]         2       Sqrt[2]          2
                2 w                      2 w

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