# Some help with a differentiation of vector components with respect to themselves

Lets say that we have a vector with 3 components $n_1, n_2, n_3$, each of which depends on 3 coordinates $x_1, x_2, x_3$:

$$\textbf{n}=(n_1(x_1,x_2,x_3), n_2(x_1, x_2, x_3), n_3(x_1, x_2, x_3))$$

By comma, let's symbolize differentiation with respect to the index that follows, i.e., $$n_{1,1}=\partial{n_{1}(x_1,x_2,x_3)}/\partial{x_1}$$

What is the result for the following:
$$\partial{n_{1,2}}/\partial{n_{2,3}}$$

Edit: Consider the functions of $n_1, n_2, n_3$ to be differentiable n-times (e.g. trigonometric) to avoid constants/zeros.

• are you sure you want to ask this. this is meaningless. for example both ${n_{1,2}}$ and ${n_{2,3}}$ could be constants. since they are functions not variables, they cud as well be ... such doubts are bound to occur when you are starting ... good question though. – magguu Mar 7 '13 at 12:46

In a $d$-dimensional setting (here $d=3$) you have to select $d$ "independent" functions as your coordinate variables. Such a triple can be the cartesian coordinates $(x_1,x_2,x_3)$, spherical coordinates $(r,\phi,\theta)$, cylindrical coordinates $(\rho,\phi, z)$, or some other more general coordinates appropriate for your problem.
Only after you have selected a complete such triple you are allowed to talk about partial derivatives with respect to the coordinate variables appearing therein. The partial derivation operators, like $${\partial\over \partial r},\ {\partial\over \partial \phi},\ {\partial\over \partial \theta}$$ form the dual basis (in the sense of linear algebra) of the basis formed by the coordinate differentials $dr$, $d\phi$, $d\theta$, and are inherently linked to all three of them.
It follows that given some two functions $f,\ g:\ \Bbb R^3\to \Bbb R$ it makes absolutely no sense to talk about a standalone derivative ${\displaystyle{\partial f\over\partial g}}$. In fact the quotient of the infinitesimal changes in $f$ and $g$ under a small displacement starting at a point $p$ depends heavily on the direction of this displacement.
• Thanks for the reply! Let's say I choose $(x_1, x_2,x_3)$ and I chose 1 specific direction $x_1$. Is it then that the result will be $n_{1,21}/n_{2,31}$ – noob Mar 7 '13 at 14:28
• @noob: As you indicate, the quotient $n_{1,21}/n_{2,31}$ has a definite meaning. It describes the ratio of the infinitesimal changes of the quantities $n_{1,2}$ and $n_{2,3}$ under a displacement in the $x_1$-direction. – Christian Blatter Mar 7 '13 at 16:19