$F\left(u\,v\right)\equiv f\left(x\left(u\,v\right)\,y\left(u\,v\right)\,z\left(u\,v\right)\right)$

$∥\mathbf{N}∥\equiv \sqrt{{\left(\frac{\partial \left(y\,z\right)}{\partial \left(u\,v\right)}\right)}^2\+{\left(\frac{\partial \left(z\,x\right)}{\partial \left(u\,v\right)}\right)}^2\+{\left(\frac{\partial \left(x\,y\right)}{\partial \left(u\,v\right)}\right)}^2}$

$\int {\int }_{S}f \mathit{\ⅆ}\mathrm{\σ} \={\int }_{u\=a}^{u\=b}{\int }_{v\=g\left(u\right)}^{v\=h\left(u\right)}F\left(u\mathit{\,}v\right)∥\mathbf{N}∥ \mathit{\ⅆ}v \mathit{\ⅆ}u$

$\=$

$b\=$

$\int {\int }_{S}f \mathit{\ⅆ}\mathrm{\σ}\=$${\int }_{v\=a}^{v\=b}{\int }_{u\=G\left(v\right)}^{u\=H\left(v\right)}F\left(u\mathit{\,}v\right)∥\mathbf{N}∥ \mathit{\ⅆ}u \mathit{\ⅆ}v$

$\=$

$a\=$