Hi I'm just looking at surface integrals and I keep coming across this formula.
$$\textrm{d}S=\frac{\textrm{d}x\textrm{d}y}{\hat{n} \cdot \hat{k}}$$
So I believe this is just saying some differential of the surface is equal to some differential of area divided by some factor which is the dot product of two vectors.
Can anyone give me an intuition as to where this comes from so I know what it does and how to use it?
I'm only familiar with $$\textrm{d}S=\left| \frac{\partial f}{\partial x} \times \frac{\partial f}{\partial y}\right|\textrm{d}x\textrm{d}y$$
$\hat{\mathbf{k}}$ is the "up" vector, and $\hat{\mathbf{n}}$ is the normal.
The dot product $\hat{\mathbf{k}} \cdot \hat{\mathbf{n}} = \cos\theta$, where $\theta$ is the angle between $\hat{\mathbf{k}}$ and $\hat{\mathbf{n}}$.
If the angle is not $0$, then the rectangle created by $dx$ and $dy$ is not parallel to the surface. Hence the surface "patch" $dS$ is a little larger than $dx\,dy$. That is why there is a scaling factor $1/\hat{\mathbf{k}} \cdot \hat{\mathbf{n}}$