I want expression for Laplacian in orthogonal curvilinear coordinates ($u,v$). I can obtain the expression by two methods and they seem to give different results. Can anybody point out the mistake?
\begin{eqnarray} \frac{\partial }{\partial x}\frac{\partial p}{\partial x} &=& \frac{\partial }{\partial x}\left(u_x p_u + v_x p_v\right) \\ &=& u_{xx}p_u + u_xp_{ux} + v_{xx}p_v + v_xp_{vx} \\ &=& u_{xx}p_u + u_x\partial_{u}(u_x p_u + v_x p_v) + v_{xx}p_v + v_x \partial_{v}(u_x p_u + v_x p_v) \\ &=& u_{xx}p_u + u^2_xp_{uu}+ u_xv_xp_{uv} + v_{xx}p_v + v^2_xp_{vv} + u_xv_xp_{uv}\\ &=& u_{xx}p_u + u^2_xp_{uu} + v_{xx}p_v + v^2_xp_{vv} + 2 u_xv_xp_{uv} \\ \end{eqnarray}
similarly, \begin{eqnarray} \frac{\partial }{\partial y}\frac{\partial p}{\partial y} &=& u_{yy}p_r + u^2_yp_{uu} + v_{yy}p_s + v^2_yp_{vv} + 2 u_yv_yp_{uv} \end{eqnarray}
So that, \begin{eqnarray} \frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2} &=& (u_{xx} + u_{yy} )p_u + (u^2_y + u^2_x)p_{uu} + (v_{xx}+v_{yy})p_v + (v^2_x+v^2_y)p_{vv} + 2 (r_ys_y + r_xs_x)u_{uv} \\ \nabla ^2 p &=& (u_{xx} + u_{yy} )p_u + (u^2_y + u^2_x)p_{uu} + (v_{xx}+v_{yy})p_v + (v^2_x+v^2_y)p_{vv} \\ \nabla ^2 p &=& (\Delta u)p_u + (|\nabla u|)^2 p_{uu} + (\Delta v)p_v + (|\nabla v|)^2 p_{vv} \quad{Eq:first} \end{eqnarray}
Expression for Laplacian can also be obtained via, \begin{eqnarray} \nabla = \left(\frac{1}{h_u}\frac{\partial }{\partial u},\frac{1}{h_v}\frac{\partial }{\partial v}\right) \end{eqnarray}
where, \begin{eqnarray} h_u = \left|\frac{\partial \vec r}{\partial u}\right| \quad \quad h_v = \left|\frac{\partial \vec r}{\partial v}\right| \end{eqnarray}
\begin{eqnarray} \nabla ^2 p &=& \frac{1}{h_u h_v}\left(\frac{\partial }{\partial u}\left[\frac{h_v}{h_u}\frac{\partial p}{\partial u}\right] + \frac{\partial }{\partial v}\left[\frac{h_u}{h_v}\frac{\partial p}{\partial v}\right]\right) \quad{Eq:second} \end{eqnarray}
When I want to compare the two results Eq.({Eq:first}) and Eq.({Eq:second}), coefficients of the term $p_{uu}$ are different.
\begin{eqnarray} |\nabla u|^2 &=&\left|\left(\frac{\partial u}{\partial x}\right)^2 +\left(\frac{\partial u}{\partial y}\right)^2 \right| \\ \frac{1}{h^2_u} &=& \frac{1}{\left(\frac{\partial x}{\partial u}\right)^2 +\left(\frac{\partial y}{\partial u}\right)^2} \end{eqnarray}
These coefficients are clearly different. Can one show that exp(1) and exp(2) are the same? If yes, what mistake am I making?