I am trying that in order to calculate the volume under a graph of a function $\phi : \mathbb{R}^k \to \mathbb{R}^{k+1}$ the Area element we should integrate is $$ \sqrt{1+\left|\nabla\phi\left(x\right)\right|^{2}} $$ (over the set we want to calcualte the volume on).
All I have left to do right now is to calculate the determinant of $$ \begin{bmatrix}1+\left(\frac{\partial\phi\left(x\right)}{\partial x_{1}}\right)^{2} & \frac{\partial\phi\left(x\right)}{\partial x_{1}}\frac{\partial\phi\left(x\right)}{\partial x_{2}} & \cdots & \frac{\partial\phi\left(x\right)}{\partial x_{1}}\frac{\partial\phi\left(x\right)}{\partial x_{k}}\\ \frac{\partial\phi\left(x\right)}{\partial x_{2}}\frac{\partial\phi\left(x\right)}{\partial x_{1}} & 1+\left(\frac{\partial\phi\left(x\right)}{\partial x_{2}}\right)^{2} & \cdots & \frac{\partial\phi\left(x\right)}{\partial x_{2}}\frac{\partial\phi\left(x\right)}{\partial x_{k}}\\ \vdots\\ \\ \frac{\partial\phi\left(x\right)}{\partial x_{k}}\frac{\partial\phi\left(x\right)}{\partial x_{1}} & \cdots & & 1+\left(\frac{\partial\phi\left(x\right)}{\partial x_{k}}\right)^{2} \end{bmatrix} $$
Which should be $$ 1+\left|\nabla\phi\left(x\right)\right|^{2} $$
And Im done.
How can I do it?
Thanks in advance.