What's taught in calculus:

In the calculus of multiple variables I learned that the Jacobian

$$\textbf J=\frac{\partial(x_1,\ldots,x_n)}{\partial(t_1,\ldots,t_n)}=\left(\begin{array}{ccc}\frac{\partial x_1}{\partial t_1}&\cdots&\frac{\partial x_1}{\partial t_n}\\\vdots&\ddots&\vdots\\\frac{\partial x_n}{\partial t_1}&\ldots&\frac{\partial x_n}{\partial t_n}\end{array}\right)$$

gives me a means of obtaining differentials $dt_1,\ldots,dt_n$ from $dx_1,\ldots,dx_n$. Geometric intuition is provided: The collection $dt$ is the volume/area element of an infinitesimal portion to be integrated. The equation for each member of $dx$ expressed as a vector is

$$d\textbf x_k=\sum_{i=0}^n\frac{\partial x_k}{\partial t_i}d\textbf t_i$$

and in a two-dimensional system, we have

$$d\textbf x_k=\left<\frac{\partial x_k}{\partial t_1}dt_1,\frac{\partial x_k}{\partial t_2}dt_2,0\right>$$

Then we find the area element with $dx$ corresponding to $dt$

$$|d\textbf x_1\times d\textbf x_2|=\left|\frac{\partial x_1}{\partial t_1}\frac{\partial x_2}{\partial t_2}-\frac{\partial x_1}{\partial t_2}\frac{\partial x_2}{\partial t_1}\right|dt_1dt_2$$

which is the same as $|\det\textbf J|=\left|\det\frac{\partial(x_1,x_2)}{\partial(t_1,t_2)}\right|$, which concludes the motivation for the Jacobian in basic calculus. I'm confident the same exercise as above could be performed in 3D, except with $d\textbf x_1\cdot(d\textbf x_2\times d\textbf x_3)$ for my volume element, but that's where it ends, since there's no notion of cross products above 3D in calculus.

My questions:

  1. Cross products don't extend to above three dimensions. So far I've had 3D Jacobians explained. What about $n$-dimensional Jacobians?
  2. What about coordinate transformations with $T:(t_1,\ldots,t_m)\to(x_1,\ldots,x_n)$ and $n\neq m$?
  3. Is the motivation given in calculus at all rigorous? I'm still not certain exactly what a differential is, and why I can manipulate them as above.
  • 1
    $\begingroup$ wedge product generalize the cross product $\endgroup$ – janmarqz Jan 24 '15 at 21:35

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