Integration by Parts in 2D: How to Apply the Rule in Polar Coordinates?

[soks]

The integration by parts rule in two dimensions is
$\int_{Ω}\frac{\partial w}{\partial x_{i}} v dΩ = \int_{\Gamma} w v \vec{n} d\Gamma - \int_{Ω} w \frac{\partial v}{\partial x_{i}} dΩ$

I have two examples in polar coordinates
In first example I have $\vec{n}=\vec{n_{r}}$

$\int_{\Gamma} \frac{1}{r^{2}} \frac{\partial^{2}w}{\partial\varphi^{2}}\frac{∂ v}{\partial r} \vec{n_{r}} d\Gamma = -2 \int_{Ω}\frac{1}{r^{3}}\frac{\partial^{2} w}{\partial \varphi^{2}} \frac{∂v}{∂r} dΩ + \int_{Ω}\frac{1}{r^{2}}\frac{\partial^{3} w}{∂r \partial \varphi^{2} } \frac{∂v}{∂r} dΩ + \int_{Ω}\frac{1}{r^{2}}\frac{\partial^{2} w}{\partial \varphi^{2} } \frac{∂^{2}v}{∂r^{2}} dΩ$

and in second $\vec{n}=\vec{n_{\varphi}}$

$\int_{\Gamma} \frac{1}{r^{2}} \frac{\partial w}{\partial\varphi}\frac{∂ v}{\partial r} \vec{n_{\varphi}} d\Gamma = \int_{Ω}\frac{1}{r^{3}}\frac{\partial^{2} w}{\partial \varphi^{2}} \frac{∂v}{∂r} dΩ + \int_{Ω}\frac{1}{r^{3}}\frac{\partial w}{\partial \varphi} \frac{∂^{2}v}{∂r∂\varphi} dΩ$

When I integrate with respect to $\varphi$ I multiply equation by $\frac{1}{r}$ but I am no sure if this is correct.

Are this two solutions correct?

Thanks for answers

 

[soks]

Functions w and v are functions of r and $\varphi$ ( w = w(r, $\varphi$) and v = v(r, $\varphi$))