How to compute Hyper-area? The function $A=(\sin(y)\sin(z)+\cos(y)\cos(z))\sin(w)\sin(x)+\cos(w)\cos(x)$, given $w\in[0,\pi], x\in[0,\pi], y\in[0,2\pi], z\in[0,2\pi]$, defines a three-dimensional "surface" in 4D.  ($A = f(w,x,y,z)$ represent level sets). How would I calculate the hyper-area of this surface as a function of A?
thank you!
p.s. I don't necessarily need a closed-form solution, I'm going to evaluate the integral numerically, but I don't know what the integral should be.
 A: Given a differentiable function $f : U \to \mathbb R^m$ where $U \subset \mathbb R^n$ is open, the $n$-dimensional content/volume of $f(U)$ is traditionally defined as the integral:
$$\mu(f(u)) := \int_U \sqrt{ Det( (f')^t f' ) }$$
where $f'$ is the derivative of $f$, thought of (point-wise) as an $m$-by-$n$ matrix. $(f')^t$ is the transpose of that matrix. 
I'll perhaps add to this response in a few moments. 
edit
Okay, so although the above reply is technically correct it's not really useful for what you need to do since you don't have a parametrization of your level-set.  At least, not yet.  But there is a nice parametrization.  There's also a pretty cute way to compute the volume without computing a parametrization, maybe I'll describe that later. 
Your level set has a lot of symmetry and we can make use of that.  Think of $x$, $y$, $z$ and $w$ as being real numbers.  Then there is a symmetry $(x,y,z,w) \longmapsto (x,y+a,z+a,w)$ where $a$ is an arbitrary real number.  
Since $\sin y \sin z + \cos y \cos z$ is a number between $-1$ and $1$, call it $\alpha$.  Similarly, let $\beta = \sin w \sin x + \cos w \cos x$, then this turns the problem into one of solving for
$$\alpha \beta + (1-\alpha) \cos w \cos x = A$$
You can solve for $\beta$, giving
$$\beta = c - \frac{1-\alpha}{\alpha} \cos w \cos x $$
So now you have a parametrization of your domain.  Perhaps it's not so clean, especially since  the domain of integration is described by all solutions to $-1 \leq c-\frac{1-\alpha}{\alpha} \cos w \cos x \leq 1$.  
But if you have a software package where you can specify the domain of integration for a multi-variable function in terms of functional inequalities, this should be fine.  Hmm, maybe there's a better way to approach this problem.  
** 2nd edit **
And the "cute" way to compute the area would be with the appropriate Cauchy-Crofton theorem.   The corresponding theorem for lengths of planar curves is here: http://en.wikipedia.org/wiki/Crofton_formula but there are analogous formula for $n$-dimensional content of submanifolds of $\mathbb R^m$.  The integral is:
$$ \int_{E_{m,n}} |L \cap N| dL $$
here $N \subset \mathbb R^m$ is an $n$-dimensional submanifold of $\mathbb R^n$.  $E_{m,n}$ is the space of $(m-n)$-dimensional affine-linear subspaces of $\mathbb R^m$ given a certain natural measure.  Given $L \in E_{m,n}$, $L \cap N$ is the intersection of $L$ with $N$, and $|L\cap N| \in [0,\infty]$ is the cardinality of the intersection.  The space where $|L \cap N|=\infty$ has measure zero in $E_{m,n}$ so that's not relevant to the integration.  Technically there is a constant in front of the above integral and it's generally reasonable to work out.   In your case, $m=4$, $n=3$ and $E_{4,3}$ is the tangent bundle of $\mathbb RP^3$ so it has a fairly natural parametrization by $S^3 \times \mathbb R^3$.  Approximating the above integral is computationally a little less complicated than rendering (ray-tracing) a small scene. 
