${}$
The positionvector form for the parametrically given surface is
$\mathbf{R}\=x\left(u\,v\right)\mathbf{i}plus;y\left(ucomma;v\right)\mathbf{j}plus;z\left(ucomma;v\right)\mathbf{k}$
${}$
Represented as position vectors, the following are coordinate curves on the surface along which $v\=b$ and $u\=a$, respectively.
${}$
${\mathbf{R}}_{u}\=x\left(u\,b\right)\mathbf{i}plus;y\left(ucomma;b\right)\mathbf{j}plus;z\left(ucomma;b\right)\mathbf{k}$ and ${\mathbf{R}}_{v}\=x\left(a\,v\right)\mathbf{i}plus;y\left(acomma;v\right)\mathbf{j}plus;z\left(acomma;v\right)\mathbf{k}$
${}$
Vectors tangent to these curves are
${\mathbf{T}}_{u}\={x}_{u}\mathbf{i}plus;{y}_{u}\mathbf{j}plus;{z}_{u}\mathbf{k}$ and ${\mathbf{T}}_{v}\={x}_{v}\mathbf{i}plus;{y}_{v}\mathbf{j}plus;{z}_{v}\mathbf{k}$
A vector normal to the surface at $\left(a\,b\right)$ is then $\mathbf{N}\={\mathbf{T}}_{u}\times {\mathbf{T}}_{v}$, that is,
$\mathbf{N}$

$\=verbar;\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ {x}_{u}& {y}_{u}& {z}_{u}\\ {x}_{v}& {y}_{v}& {z}_{v}\end{array}verbar;$ ${}$

$\=\left\begin{array}{cc}{y}_{u}& {z}_{u}\\ {y}_{v}& {z}_{v}\end{array}verbar;\mathbf{i}verbar;\begin{array}{cc}{x}_{u}& {z}_{u}\\ {x}_{v}& {z}_{v}\end{array}\right\mathbf{j}plus;verbar;\begin{array}{cc}{x}_{u}& {y}_{u}\\ {x}_{v}& {y}_{v}\end{array}verbar;\mathbf{k}$

${}$

${}$

$\=\left\begin{array}{cc}{y}_{u}& {y}_{v}\\ {z}_{u}& {z}_{v}\end{array}verbar;\mathbf{i}verbar;\begin{array}{cc}{x}_{u}& {x}_{v}\\ {z}_{u}& {z}_{v}\end{array}\right\mathbf{j}plus;verbar;\begin{array}{cc}{x}_{u}& {x}_{v}\\ {y}_{u}& {y}_{v}\end{array}verbar;\mathbf{k}$

${}$

${}$

$\=\left\begin{array}{cc}{y}_{u}& {y}_{v}\\ {z}_{u}& {z}_{v}\end{array}verbar;\mathbf{i}plus;verbar;\begin{array}{cc}{z}_{u}& {z}_{v}\\ {x}_{u}& {x}_{v}\end{array}\right\mathbf{j}plus;verbar;\begin{array}{cc}{x}_{u}& {x}_{v}\\ {y}_{u}& {y}_{v}\end{array}verbar;\mathbf{k}$

${}$

${}$

$\=\frac{\partial \left(ycomma;z\right)}{\partial \left(ucomma;v\right)}\mathbf{i}plus;\frac{\partial \left(zcomma;x\right)}{\partial \left(ucomma;v\right)}\mathbf{j}plus;\frac{\partial \left(xcomma;y\right)}{\partial \left(ucomma;v\right)}\mathbf{k}$

${}$

${}$

$\={J}_{1}\mathbf{i}plus;{J}_{2}\mathbf{j}plus;{J}_{3}\mathbf{k}$



where the determinant representing the cross product has been firstrow 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 $\u2225\mathbf{N}\u2225\=\sqrt{{J}_{1}^{2}\+{J}_{2}^{2}\+{J}_{3}^{2}}$ = $\mathrm{\λ}$, so $\mathrm{d\σ}\=\mathrm{\λ}d\stackrel{Hat;}{A}$.
${}$