The position-vector form for the parametrically given surface is

$\mathbf{R}\=x\left(u\,v\right) \mathbf{i}\+y\left(u\,v\right) \mathbf{j}\+z\left(u\,v\right) \mathbf{k}$ 

Represented as position vectors, the following are coordinate curves on the surface along which $v\=b$ and $u\=a$, respectively.

${\mathbf{R}}_u\=x\left(u\,b\right) \mathbf{i}\+y\left(u\,b\right) \mathbf{j}\+z\left(u\,b\right) \mathbf{k}$   and    ${\mathbf{R}}_v\=x\left(a\,v\right) \mathbf{i}\+y\left(a\,v\right) \mathbf{j}\+z\left(a\,v\right) \mathbf{k}$ 

Vectors tangent to these curves are

${\mathbf{T}}_u\=x_u \mathbf{i}\+y_u \mathbf{j}\+z_u \mathbf{k}$   and   ${\mathbf{T}}_v\=x_v \mathbf{i}\+y_v \mathbf{j}\+z_v \mathbf{k}$ 

A vector normal to the surface at $\left(a\,b\right)$ is then $\mathbf{N}\={\mathbf{T}}_u\times {\mathbf{T}}_v$, that is,

where the determinant representing the cross product has been first-row expanded. In the second row of the display, the property that a determinant does not change if the array is transposed is used. In the third row of the display, the property that a determinant changes sign if two rows are interchanged is used.

The length of N is then $∥\mathbf{N}∥\=\sqrt{J_1^2\+J_2^2\+J_3^2}$ = $\mathrm{\λ}$, so $\mathrm{d\σ}\=\mathrm{\λ} d\stackrel{\^}{A}$.