Fdtxjr

Define $varphi:(0,infty)times S^ntomathbbR^n+1$ by $varphi(t,theta)=ttheta$. Let $E$ be a subset of $mathbbR^n+1$ and let $E_rsubseteqmathbbR^n+1$ contain all points $thetain S^n$ such that $tthetain E$ for some $t>0$. Then $int_Efdx_1...dx_n+1=int_-infty^inftyint_E_rfcircvarphi r^ndrdv$, where $dv$ is the volume form of the n sphere.

I'm with the classical change of variable. For this, I need to know why the determinant of $Dvarphi=r^n$, but I don't see an easy way of seeing this.

Now that the meaning of $E_r$ is cleared up, I can say with certainty that your iterated integral formula makes no sense. There is a small problem that the integrals are backwards from the differentials. ($r$ does not vary over $E_r$, nor does $v$ vary over $(-infty, infty)$.)

But the real issue is that the region of integration for iterated integral does not match $E$ at all. To give the correct expression, you need to define another collection of sets $$E(theta) = t in Bbb R : ttheta in E, t > 0quadforall theta in E_r$$
(I didn't want to use $E_theta$ because $E(theta)$ depends on $theta$, while the previously established "$E_r$" does not depend on $r$.) Then the actual formula would be
$$int_EfdV=int_E_rint_E(theta)fcircvarphi r^n,dr,dv$$

If you turned it around: Let $E_r subseteq S^n$ and let $E = ttheta : theta in E_r, t > 0$ (i.e., $E$ is the union of all rays passing through $E_r$), then you could justify the equation
$$int_EfdV=int_0^inftyint_E_rfcircvarphi r^n,dv,dr$$
since $E(theta) = (0,infty)$ for all $theta$ in this case. But there is no way to justify integrating $r$ over $(-infty, infty)$.

As for your differential question, informally: $dr$ is the direction perpendicular to $rS^n$, the sphere of radius $r$, so the volume element at radius $r$ is $dr$ times the "$n$-area" $dw$ on $rS^n$. But $rS^n$ is just $S^n$ magnified by a factor of $r$, so every one of its $n$ independent directions is scaled by $r$, and therefore, its differential area is $r^n$ times as big as the corresponding differential area $dv$ on $S^n$.

To make that handwaving rigorous, define an orthonormal coordinate system on $S^n$. For instance, $(theta_1, ..., theta_n)$, where $$x_0 = sintheta_1 ... sintheta_n\x_1 = costheta_1sin theta_2 ... sin theta_n\x_2 = costheta_2sin theta_3 ... sin theta_n\vdots\x_n = cos theta_n$$
And therefore $$varphi(r,theta_1, ..., theta_n) = (rsintheta_1 ... sintheta_n, rcostheta_1sin theta_2 ... sin theta_n, ..., rcostheta_n)$$
Now you have something concrete to base your Hessian calculation on.

I prefer to see this in terms of metric tensors; $$tag1ds^2=dr^2 + r^2dtheta^2, $$where $ds^2$ is the metric tensor of $mathbb R^n+1$, and $dtheta^2$ is the metric tensor of $mathbb S^n$. This formula can be quickly proved by expanding $ds^2=(d(rtheta))^2=(dr, theta+r,dtheta)^2, $ using that $thetacdot dtheta=0$, because $thetacdot theta=1$.

Once (1) has been proved, the volume form formula immediately follows from $$dx=sqrtgdrdtheta, $$ where $g$ is the determinant of $ds^2$.

Not too rigorous, maybe, but faster.