Gradient and Divergence in Orthonormal Curvilinear Coordinates

Gradient and Divergence in Orthonormal Curvilinear CoordinatesSwapnil Sunil JainAug 7, 2006

Gradient and Divergence in Orthonormal Curvilinear Coordinates

Gradient in Curvilinear Coordinates

In rectangular coordinates (where $f=f(x,y,z)$), an infinitesimal length vector $d\overrightarrow{l}$ is given by

$d\overrightarrow{l}=dx\widehat{x}+dy\widehat{y}+dz\widehat{z}$

the gradient is given by

$\nabla ={\displaystyle \frac{\partial }{\partial x}}\widehat{x}+{\displaystyle \frac{\partial }{\partial y}}\widehat{y}+{\displaystyle \frac{\partial }{\partial z}}\widehat{z}$

and the differential change in the output is given by

$df=\nabla f\circ d\overrightarrow{l}={\displaystyle \frac{\partial f}{\partial x}}dx+{\displaystyle \frac{\partial f}{\partial y}}dy+{\displaystyle \frac{\partial f}{\partial z}}dz$

Similarly in orthonormal curvilinear coordinates ( where $f=f(q_1,q_2,q_3)$), the infinitesimal length vector is given by

$d\overrightarrow{l}=h_{1}d{q}_1{\widehat{q}}_1+h_{2}d{q}_2{\widehat{q}}_2+h_{3}d{q}_3{\widehat{q}}_3$

where

$h_i=\sqrt{{\displaystyle \sum _k}{\left({\displaystyle \frac{\partial {\overrightarrow{x}}_k}{\partial q_i}}\right)}^2}\text{ and }{\widehat{q}}_i={\displaystyle \frac{1}{h_i}}\left({\displaystyle \frac{\partial {\overrightarrow{x}}_k}{\partial q_i}}\right)\text{ for }i\in 1,2,3$

So if

$\nabla =\alpha {\displaystyle \frac{\partial }{\partial q_1}}{\widehat{q}}_1+\beta {\displaystyle \frac{\partial }{\partial q_2}}{\widehat{q}}_2+\gamma {\displaystyle \frac{\partial }{\partial q_3}}{\widehat{q}}_3$

then since we know that

$df={\displaystyle \frac{\partial f}{\partial q_1}}d{q}_1+{\displaystyle \frac{\partial f}{\partial q_2}}d{q}_2+{\displaystyle \frac{\partial f}{\partial q_3}}d{q}_3$

and

$df=\nabla F\circ d\overrightarrow{l}=\alpha h_1{\displaystyle \frac{\partial f}{\partial q_1}}d{q}_1+\beta h_2{\displaystyle \frac{\partial f}{\partial q_2}}d{q}_2+\gamma h_3{\displaystyle \frac{\partial f}{\partial q_3}}d{q}_3$

this implies that

$\alpha ={\displaystyle \frac{1}{h_i}};\beta ={\displaystyle \frac{1}{h_2}};\gamma ={\displaystyle \frac{1}{h_3}}$

Hence,

$\nabla$$=$${\displaystyle \frac{1}{h_1}}{\displaystyle \frac{\partial }{\partial q_1}}{\widehat{q}}_1+{\displaystyle \frac{1}{h_2}}{\displaystyle \frac{\partial }{\partial q_2}}{\widehat{q}}_2+{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{\partial }{\partial q_3}}{\widehat{q}}_3$

$=$${\displaystyle \sum _i}{\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial }{\partial q_i}}{\widehat{q}}_i$

Divergence in Curvilinear Coordinates

In the previous section we concluded that in curvilinear coordinates, the gradient operator $\nabla$ is given by

$\nabla ={\displaystyle \sum _i}{\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial }{\partial q_i}}{\widehat{q}}_i$

Then for $\overrightarrow{F}=F_1{\widehat{q}}_1+F_2{\widehat{q}}_2+F_3{\widehat{q}}_3$, the divergence of $\overrightarrow{F}$ is given by

$\nabla \circ \overrightarrow{F}=\left({\displaystyle \sum _i}{\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial }{\partial q_i}}{\widehat{q}}_i\right)\circ \overrightarrow{F}$

which is not equal to

$\left({\displaystyle \sum _i}{\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial }{\partial q_i}}{\widehat{q}}_i\right)\circ \overrightarrow{F}\ne {\displaystyle \sum _i}{\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial F_i}{\partial q_i}}$

as one would think! The real expression can be derived the following way,

$={\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial }{\partial q_i}}{\widehat{q}}_i\right)\circ \overrightarrow{F}\right]$

$={\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ \left({\displaystyle \frac{\partial \overrightarrow{F}}{\partial q_i}}\right)\right]$

$={\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ \left({\displaystyle \frac{\partial }{\partial q_i}}\left({\displaystyle \sum _j}{F}_j{\widehat{q}}_j\right)\right)\right]$

$={\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ \left({\displaystyle \sum _j}{\displaystyle \frac{\partial }{\partial q_i}}\left(F_j{\widehat{q}}_j\right)\right)\right]$

$={\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ \left({\displaystyle \sum _j}{\widehat{q}}_j{\displaystyle \frac{\partial F_j}{\partial q_i}}+F_j{\displaystyle \frac{\partial {\widehat{q}}_j}{\partial q_i}}\right)\right]$

$={\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ \left({\displaystyle \sum _j}{\widehat{q}}_j{\displaystyle \frac{\partial F_j}{\partial q_i}}+{\displaystyle \sum _j}{F}_j{\displaystyle \frac{\partial {\widehat{q}}_j}{\partial q_i}}\right)\right]$

$={\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ {\displaystyle \sum _j}{\widehat{q}}_j{\displaystyle \frac{\partial F_j}{\partial q_i}}+\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ {\displaystyle \sum _j}{F}_j{\displaystyle \frac{\partial {\widehat{q}}_j}{\partial q_i}}\right]$

$=\underset{\text{call it A}}{\underbrace{{\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ {\displaystyle \sum _j}{\widehat{q}}_j{\displaystyle \frac{\partial F_j}{\partial q_i}}\right]}}+\underset{\text{call it B}}{\underbrace{{\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ {\displaystyle \sum _j}{F}_j{\displaystyle \frac{\partial {\widehat{q}}_j}{\partial q_i}}\right]}}$

$A$$=$${\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ {\displaystyle \sum _j}{\widehat{q}}_j{\displaystyle \frac{\partial F_j}{\partial q_i}}\right]$

$=$${\displaystyle \sum _i}\left[{\displaystyle \frac{1}{h_i}}{\displaystyle \sum _j}\underset{{\delta }_{ij}}{\underbrace{({\widehat{q}}_i\circ {\widehat{q}}_j)}}{\displaystyle \frac{\partial F_j}{\partial q_i}}\right]$

$=$${\displaystyle \sum _i}{\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial F_i}{\partial q_i}}$

$B$$=$${\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ {\displaystyle \sum _j}{F}_j{\displaystyle \frac{\partial {\widehat{q}}_j}{\partial q_i}}\right]$

Using the following equality

${\displaystyle \frac{\partial {\widehat{q}}_j}{\partial q_i}}={\displaystyle \frac{{\widehat{q}}_i}{h_j}}{\displaystyle \frac{\partial h_i}{\partial q_j}}\mathit{\hspace{1em}\hspace{1em}}\forall i\ne j$

we can write $B$ as

$B$$=$${\displaystyle \sum _i}\left[\left({\displaystyle \frac{1}{h_i}}{\widehat{q}}_i\right)\circ {\displaystyle \sum _j}{F}_j\left({\widehat{q}}_i{\displaystyle \frac{1}{h_j}}{\displaystyle \frac{\partial h_i}{\partial q_j}}\right)\right]\mathit{\hspace{1em}\hspace{1em}}\forall i\ne j$

$=$${\displaystyle \sum _i}\left[{\displaystyle \frac{1}{h_i}}{\displaystyle \sum _j}{F}_j\underset{1}{\underbrace{({\widehat{q}}_i\circ {\widehat{q}}_i)}}{\displaystyle \frac{1}{h_j}}{\displaystyle \frac{\partial h_i}{\partial q_j}}\right]\mathit{\hspace{1em}\hspace{1em}}\forall i\ne j$

$=$${\displaystyle \sum _i}\left[{\displaystyle \frac{1}{h_i}}{\displaystyle \sum _j}{F}_j{\displaystyle \frac{1}{h_j}}{\displaystyle \frac{\partial h_i}{\partial q_j}}\right]\mathit{\hspace{1em}\hspace{1em}}\forall i\ne j$

$=$${\displaystyle \sum _{i\ne j}}{\displaystyle \frac{F_j}{h_j{h}_i}}{\displaystyle \frac{\partial h_i}{\partial q_j}}$

$=$${\displaystyle \sum _{i\ne 1}}{\displaystyle \frac{F_1}{h_1{h}_i}}{\displaystyle \frac{\partial h_i}{\partial q_1}}+{\displaystyle \sum _{i\ne 2}}{\displaystyle \frac{F_2}{h_2{h}_i}}{\displaystyle \frac{\partial h_i}{\partial q_2}}+{\displaystyle \sum _{i\ne 3}}{\displaystyle \frac{F_3}{h_3{h}_i}}{\displaystyle \frac{\partial h_i}{\partial q_3}}$

$\Rightarrow \nabla \circ \overrightarrow{F}$$=$$A+B$

$=$${\displaystyle \sum _i}{\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial F_i}{\partial q_i}}+{\displaystyle \sum _{i\ne 1}}{\displaystyle \frac{F_1}{h_1{h}_i}}{\displaystyle \frac{\partial h_i}{\partial q_1}}+{\displaystyle \sum _{i\ne 2}}{\displaystyle \frac{F_2}{h_2{h}_i}}{\displaystyle \frac{\partial h_i}{\partial q_2}}+{\displaystyle \sum _{i\ne 3}}{\displaystyle \frac{F_3}{h_3{h}_i}}{\displaystyle \frac{\partial h_i}{\partial q_3}}$

$=$$\left[{\displaystyle \frac{1}{h_1}}{\displaystyle \frac{\partial F_1}{\partial q_1}}+{\displaystyle \frac{1}{h_2}}{\displaystyle \frac{\partial F_2}{\partial q_2}}+{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{\partial F_3}{\partial q_3}}\right]$

$+\left[{\displaystyle \frac{F_1}{h_1{h}_2}}{\displaystyle \frac{\partial h_2}{\partial q_1}}+{\displaystyle \frac{F_1}{h_1{h}_3}}{\displaystyle \frac{\partial h_3}{\partial q_1}}\right]$

$+\left[{\displaystyle \frac{F_2}{h_2{h}_1}}{\displaystyle \frac{\partial h_1}{\partial q_2}}+{\displaystyle \frac{F_2}{h_2{h}_3}}{\displaystyle \frac{\partial h_3}{\partial q_2}}\right]$

$+\left[{\displaystyle \frac{F_3}{h_3{h}_1}}{\displaystyle \frac{\partial h_1}{\partial q_3}}+{\displaystyle \frac{F_3}{h_3{h}_2}}{\displaystyle \frac{\partial h_2}{\partial q_3}}\right]$

Collecting similar terms together we get,

$\nabla \circ \overrightarrow{F}$$=$$\left[{\displaystyle \frac{1}{h_1}}{\displaystyle \frac{\partial F_1}{\partial q_1}}+{\displaystyle \frac{F_1}{h_1{h}_2}}{\displaystyle \frac{\partial h_2}{\partial q_1}}+{\displaystyle \frac{F_1}{h_1{h}_3}}{\displaystyle \frac{\partial h_3}{\partial q_1}}\right]$

$+\left[{\displaystyle \frac{1}{h_2}}{\displaystyle \frac{\partial F_2}{\partial q_2}}+{\displaystyle \frac{F_2}{h_2{h}_1}}{\displaystyle \frac{\partial h_1}{\partial q_2}}+{\displaystyle \frac{F_2}{h_2{h}_3}}{\displaystyle \frac{\partial h_3}{\partial q_2}}\right]$

$+\left[{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{\partial F_3}{\partial q_3}}+{\displaystyle \frac{F_3}{h_3{h}_1}}{\displaystyle \frac{\partial h_1}{\partial q_3}}+{\displaystyle \frac{F_3}{h_3{h}_2}}{\displaystyle \frac{\partial h_2}{\partial q_3}}\right]$

If we define $\mathrm{\Omega }\equiv {\mathrm{\Pi }}_i{h}_i$, we can further write the above expression as

$\nabla \circ \overrightarrow{F}$$=$$\left[{\displaystyle \frac{h_2{h}_3}{\mathrm{\Omega }}}{\displaystyle \frac{\partial F_1}{\partial q_1}}+{\displaystyle \frac{F_1{h}_3}{\mathrm{\Omega }}}{\displaystyle \frac{\partial h_2}{\partial q_1}}+{\displaystyle \frac{F_1{h}_2}{\mathrm{\Omega }}}{\displaystyle \frac{\partial h_3}{\partial q_1}}\right]$

$+\left[{\displaystyle \frac{h_1{h}_3}{\mathrm{\Omega }}}{\displaystyle \frac{\partial F_2}{\partial q_2}}+{\displaystyle \frac{F_2{h}_3}{\mathrm{\Omega }}}{\displaystyle \frac{\partial h_1}{\partial q_2}}+{\displaystyle \frac{h_1{F}_2}{\mathrm{\Omega }}}{\displaystyle \frac{\partial h_3}{\partial q_2}}\right]$

$+\left[{\displaystyle \frac{h_1{h}_2}{\mathrm{\Omega }}}{\displaystyle \frac{\partial F_3}{\partial q_3}}+{\displaystyle \frac{h_2{F}_3}{\mathrm{\Omega }}}{\displaystyle \frac{\partial h_1}{\partial q_3}}+{\displaystyle \frac{h_1{F}_3}{\mathrm{\Omega }}}{\displaystyle \frac{\partial h_2}{\partial q_3}}\right]$

$=$${\displaystyle \frac{1}{\mathrm{\Omega }}}([{\displaystyle \frac{\partial F_1}{\partial q_1}}{h}_2{h}_3+F_1{h}_3{\displaystyle \frac{\partial h_2}{\partial q_1}}+F_1{h}_2{\displaystyle \frac{\partial h_3}{\partial q_1}}]$

$+\left[h_1{\displaystyle \frac{\partial F_2}{\partial q_2}}{h}_3+{\displaystyle \frac{\partial h_1}{\partial q_2}}{F}_2{h}_3+h_1{F}_2{\displaystyle \frac{\partial h_3}{\partial q_2}}\right]$

$+[h_1{h}_2{\displaystyle \frac{\partial F_3}{\partial q_3}}+{\displaystyle \frac{\partial h_1}{\partial q_3}}{h}_2{F}_3+h_1{\displaystyle \frac{\partial h_2}{\partial q_3}}{F}_3])$

$=$${\displaystyle \frac{1}{\mathrm{\Omega }}}\left({\displaystyle \frac{\partial }{\partial q_1}}(F_1{h}_2{h}_3)+{\displaystyle \frac{\partial }{\partial q_2}}(h_1{F}_2{h}_3)+{\displaystyle \frac{\partial }{\partial q_3}}(h_1{h}_2{F}_3)\right)$

Hence,

$\nabla \circ \overrightarrow{F}$$=$${\displaystyle \frac{1}{\mathrm{\Omega }}}{\displaystyle \sum _i}{\displaystyle \frac{\partial }{\partial q_i}}\left({\displaystyle \frac{\mathrm{\Omega }}{h_i}}{F}_i\right)\mathit{\hspace{1em}}\text{ where }\mathrm{\Omega }={\mathrm{\Pi }}_i{h}_i$