\[u,v\]
so that \[\mathbf{r} = x(u,v) \mathbf{i} + y(u,v) \mathbf{j} + z(u,v) \mathbf{k}\]
is\[A =\int_{uv} | \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial u} |dudv \]
We can take
\[x,y\]
as parameters, so that \[\mathbf{r} =x \mathbf{i} + y \mathbf{j} + z(x,y) \mathbf{k}\]
\[\frac{\mathbf{r}}{\partial x} = \mathbf{i} + \frac{ \partial z}{\partial x} \mathbf{k} \]
\[\frac{\mathbf{r}}{\partial y} = \mathbf{j} + \frac{\partial z}{\partial y} \mathbf{k} \]
\[\begin{equation} \begin{aligned} | \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial u} | &=(\mathbf{i} + \frac{z}{\partial x} \mathbf{k} ) \times (\mathbf{j} + \frac{z}{\partial y} \mathbf{k}) \\ &=\left| \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & \frac{ \partial z}{\partial x} \\ 0 & 1 & \frac{ \partial z}{\partial y} \end{array} \right| \right| \\ &=| -\frac{\partial f}{\partial x} \mathbf{i} - -\frac{\partial f}{\partial x} \mathbf{j} + \mathbf{k}| \\ &= \sqrt {(\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2 +1} \end{aligned} \end{equation}\]
Hence
\[A= \int_{xy} \sqrt {(\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2 +1} dxdy \]