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I am having a really hard time in understanding the difference between coordinate and non-coordinate basis. I tried looking for some relevant questions here but none answers my concerns, may be my question is too basic.

The way I explained myself the difference between co-ordinate and non-coordinate basis is in terms of the orthonormality of the basis vectors (I am reading a text on General Relativity by Bernard Schutz). I had understood that the difference is orthonormality i.e. coordinate basis are orthonormal while non-coordinate basis are just orthogonal. But that later the later part of the text contradicts my understanding by stating that "In textbooks that deal with vector calculus in curvilinear coordinates, almost all use the unit orthonormal basis rather than coordinate basis.".

So now I am completely lost. Can anyone explain the difference between coordinate and non-coordinate basis in less mathematical terms (more in terms of the Physics or with geometric arguments that I can visualize)? Or just point me to some text I can read.

Thanks.

Update: The answer given to this question confirms my understanding. I later came to realize that the author meant that the basis vectors are called "unit orthonormal basis rather than coordinate basis".

amzon-ex
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Shaz
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  • To me the phrase 'non-coordinate basis' looks oxymoron. Does it have maybe a definition? – Berci May 27 '18 at 08:36
  • A finite-dimensional real vector space $V$ is isomorphic to ${\Bbb R}^n$. May it be that a basis in $V$ is called non-coordinate, and a basis in ${\Bbb R}^n$ is called coordinate basis in this book? Just guessing. – A.Γ. May 27 '18 at 09:08
  • So you think my understanding of the words in terms of orthogonality and orthonormality is correct? Meaning that coordinate basis are basis vectors which are orthogonal while non-coordinate basis vectors are orthonomal? – Shaz May 27 '18 at 11:05

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You mention GR, which I assume is General Relativity, and then I guess that you're talking about coordinate systems on manifolds. At each point $p$ of a manifold, there is a tangent space $T_p$, which is an ordinary vector space, so that you can talk about a basis for it.

In principle, you can choose a basis for each $T_p$ completely independently, but that's usually not very interesting. What you want is a set of smooth vector fields $X_1,\dots,X_n$ such that at each $p$ the vectors $X_1(p),\dots,X_n(p)$ are a basis for $T_p$. Sometimes this is called a frame, but I assume that this is what is meant by a basis in the text you're reading. The vector fields in a frame may or may not form an orthonormal basis for each $T_p$.

One particular way of constructing a frame is as follows: take any coordinate system $(x^1,\dots,x^n)$ and let $X_k=\partial/\partial x_k$ for $1 \le k \le n$. (Thus, $X_k$ is tangent to curves formed by changing the value of the coordinate $x_k$ and keeping all other $x_j$ fixed.) Such a coordinate frame is usually not orthonormal, although it is orthogonal in some interesting special cases, for example spherical or cylindrical coordinates in $\mathbf{R}^3$.

Hans Lundmark
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  • Thank you so much for your reply. My confusion is cleared. Sorry for being ambiguous in the question. I didnt make the question clear enough. I also read other sources after posting this question and your reply complements those sources as well. – Shaz May 27 '18 at 15:41
  • Do all riemannian manifolds admit an atlas which produces orthonormal coordinate frames? What are the main results in this regard? thanks – l4teLearner Apr 30 '23 at 11:33
  • @l4teLearner: Orthonormal? No, you can't even get orthogonal in general. Look up isothermal coordinates. – Hans Lundmark Apr 30 '23 at 18:49