Editing Curvilinear coordinates (section)

==Vector calculus==
{{See also|Differential geometry}}

===Differential elements===

In orthogonal curvilinear coordinates, since the [[total differential]] change in '''r''' is

:<math>d\mathbf{r}=\dfrac{\partial\mathbf{r}}{\partial q^1}dq^1 + \dfrac{\partial\mathbf{r}}{\partial q^2}dq^2 + \dfrac{\partial\mathbf{r}}{\partial q^3}dq^3 = h_1 dq^1 \mathbf{b}_1 + h_2 dq^2 \mathbf{b}_2 + h_3 dq^3 \mathbf{b}_3 </math>

so scale factors are <math>h_i = \left|\frac{\partial\mathbf{r}}{\partial q^i}\right|</math>

In non-orthogonal coordinates the length of  <math>d\mathbf{r}= dq^1 \mathbf{h}_1 + dq^2 \mathbf{h}_2 + dq^3 \mathbf{h}_3 </math> is the positive square root of <math>d\mathbf{r} \cdot d\mathbf{r} = dq^i dq^j \mathbf{h}_i \cdot \mathbf{h}_j </math> (with [[Einstein summation convention]]). The six independent scalar products ''g<sub>ij</sub>''='''h'''<sub>''i''</sub>.'''h'''<sub>''j''</sub> of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine ''g<sub>ij</sub>'' are the components of the [[metric tensor]], which has only three non zero components in orthogonal coordinates: ''g''<sub>11</sub>=''h''<sub>1</sub>''h''<sub>1</sub>, ''g''<sub>22</sub>=''h''<sub>2</sub>''h''<sub>2</sub>, ''g''<sub>33</sub>=''h''<sub>3</sub>''h''<sub>3</sub>.