This is a problem in my subject Calculus.
Let $E\subset\mathbb R^3$ be a pipe and $\gamma$ is its mass centerline (or weighted centerline, I have no clue what it's called in English, sorry for my bad). It means every (plane section $D_p$ of $E$ which is perpendicular to $\gamma$ at $p$) takes $p$ as its mass center (I don't know if it is called weighted center, weighted centroid or whatever, this is what I translated from my language so).
Suppose $\gamma$ is reparametrized by arclength and $\gamma''(s)\ne0\ \ \forall s$. Let $$T=\gamma',\ \ N=\frac{1}{\|T'\|}T',\ \ B=T\times N.$$
The problem start with finding det$J_\varphi(u,v,s)$ where $$\varphi(u,v,s)=\gamma(s)+uN+vB.$$
The next requirement is to show that if the area of $D_p$ is small enough then the volume of $E$ is given by $$|E|=\intop_\gamma |D_p|\,ds.$$
I have tried to calculate the Jacobian matrix of $\varphi$, but when I wrote out the coordinates of $\varphi$, this what I got \begin{align} \varphi(u,v,s)=\begin{bmatrix}x(s)+\displaystyle\frac{u}{\|T'\|}x''(s)+\frac{v}{\|T'\|}\big(z''(s)y'(s)-z'(s)y''(s)\big) \\ y(s)+\displaystyle\frac{u}{\|T'\|}y''(s)+\frac{v}{\|T'\|}\big(x''(s)z'(s)-x'(s)z''(s)\big) \\ z(s)+\displaystyle\frac{u}{\|T'\|}z''(s)+\frac{v}{\|T'\|}\big(y''(s)x'(s)-y'(s)x''(s)\big) \end{bmatrix} \end{align} I immediately realized that maybe the Jacobian matrix shouldn't be evaluated this way. And then I got stucked finding the better way.
May you help me out ? Any help is precious. Thank you much.