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## Main Question or Discussion Point

I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors.

Quoting:

$$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat x}+r\sin\theta\sin\phi\ \mathbf{\hat y}+r\cos\theta\ \mathbf{\hat z}$$And from this we get the expressions:

\begin{equation}

\begin{split}

\mathbf{e}_r&=\frac{\partial \mathbf{r}}{\partial r}&=\sin\theta\cos\phi\ \mathbf{\hat x} + \sin\theta\sin\phi\ \mathbf{\hat y} + \cos\theta\ \mathbf{\hat z} \\

\mathbf{e}_{\theta}&=\frac{\partial \mathbf{r}}{\partial \theta}&=r\cos\theta\cos\phi\ \mathbf{\hat x} + r\cos\theta\sin\phi\ \mathbf{\hat y} - r\sin\theta\ \mathbf{\hat z} \\

\mathbf{e}_{\phi}&=\frac{\partial \mathbf{r}}{\partial \phi}&=-r\sin\theta\sin\phi\ \mathbf{\hat x} + r\sin\theta\cos\phi\ \mathbf{\hat y}

\end{split}

\end{equation}Fair enough so far. I'm simultaneously reading 'A Visual Introduction to Differential Forms and Calculus on Manifolds' by Fortney. In that, a vector ##v_p## in the tangent space ##T_p(\mathbb{R}^3)## is defined as an operator acting on a real function ##f## defined on the manifold (i.e. ##f:\mathbb{R}^3\to\mathbb{R}##). The operator gives the directional derivative of ##f## in the direction ##v_p## at point ##p##:

$$v_p[f]=\sum_{i=1}^3v_i\frac{\partial f}{\partial x^i}\ \bigg|_p$$From this, we can identify the basis vectors of ##T_p(\mathbb{R}^3)## as:

$$\frac{\partial}{\partial x^1}\ \bigg|_p,\frac{\partial}{\partial x^2}\ \bigg|_p,\frac{\partial}{\partial x^3}\ \bigg|_p$$which, as you can see, are quite different from the basis vectors ##\mathbf{e}_u,\mathbf{e}_v,\mathbf{e}_w## that I defined at the start of the question.

Now you've seen that the Luscombe book formulas for ##\mathbf{e}_r,\mathbf{e}_{\theta},\mathbf{e}_{\phi}##, that I listed at the start, contain ##\mathbf{r}##. My interpretation of ##\mathbf{r}## is that it's a function from ##\mathbb{R}^3\to\mathbb{R}##, and it's a triple consisting of the three

\begin{equation}

\begin{split}

x&=x(r,\theta,\phi)&=r\sin\theta\cos\phi \\

y&=y(r,\theta,\phi)&=r\sin\theta\sin\phi \\

z&=z(r,\theta,\phi)&=r\cos\theta

\end{split}

\end{equation}are the individual coordinate functions. Since we've switched to spherical coordinates, we're now using the parameters ##r,\theta,\phi## to specify any point ##p## in the manifold ##\mathbb{R}^3##.

Using the operator definition of tangent vectors,

\begin{equation}

\begin{split}

(\mathbf{e}_r)_p[\mathbf{r}]&=v_r\frac{\partial \mathbf{r}}{\partial r}\ \bigg|_p+

v_{\theta}\frac{\partial \mathbf{r}}{\partial \theta}\ \bigg|_p+

v_{\phi}\frac{\partial \mathbf{r}}{\partial \phi}\ \bigg|_p=\frac{\partial \mathbf{r}}{\partial r}\ \bigg|_p \\

&=\frac{\partial}{\partial r}(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)\ \bigg|_p \\

&=(\sin\theta\cos\phi\,\sin\theta\sin\phi,\cos\theta)\ |_p

\end{split}

\end{equation} which matches the definition mentioned in the Luscombe book. Similarly we can calculate for the ##\theta## and ##\phi## basis vectors.

1. Does the above procedure of reconciling the two definitions for basis vectors seem correct?

2. If correct, to reconcile the definitions, I had to make a specific assumption about ##f## and act the

If you've read this far, thanks so much for the time and I'd appreciate any help!

Quoting:

The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as:Consider an arbitrary three-dimensional coordinate system where point ##P## is at the intersection of three coordinate curves labeled by ##(u,v,w)##. For a nearby point ##Q## define the vector ##\Delta \mathbf{s}\equiv\vec{PQ}##; ##\Delta \mathbf{s}## is also the vector ##\Delta \mathbf{s}=(\mathbf{r}+\Delta\mathbf{r})-\mathbf{r}##, where ##\mathbf{r}+\Delta\mathbf{r}## and ##\mathbf{r}## are the position vectors for ##Q## and ##P## relative to the origin. To first order in small quantities,$$d\mathbf{s}=\frac{\partial\mathbf{r}}{\partial u}du+\frac{\partial\mathbf{r}}{\partial v}dv+\frac{\partial\mathbf{r}}{\partial w}dw$$ where the derivatives (with respect to the coordinates) are evaluated at ##P##. The derivatives $$\mathbf{e}_u\equiv \frac{\partial \mathbf{r}}{\partial u}, \mathbf{e}_v\equiv \frac{\partial \mathbf{r}}{\partial v}, \mathbf{e}_w\equiv \frac{\partial \mathbf{r}}{\partial w}$$ form a local basis - an arbitrary ##d\mathbf{s}## in the neighborhood of ##P## can be expressed as a linear combination of them - and they're tangent to the coordinate curves.

$$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat x}+r\sin\theta\sin\phi\ \mathbf{\hat y}+r\cos\theta\ \mathbf{\hat z}$$And from this we get the expressions:

\begin{equation}

\begin{split}

\mathbf{e}_r&=\frac{\partial \mathbf{r}}{\partial r}&=\sin\theta\cos\phi\ \mathbf{\hat x} + \sin\theta\sin\phi\ \mathbf{\hat y} + \cos\theta\ \mathbf{\hat z} \\

\mathbf{e}_{\theta}&=\frac{\partial \mathbf{r}}{\partial \theta}&=r\cos\theta\cos\phi\ \mathbf{\hat x} + r\cos\theta\sin\phi\ \mathbf{\hat y} - r\sin\theta\ \mathbf{\hat z} \\

\mathbf{e}_{\phi}&=\frac{\partial \mathbf{r}}{\partial \phi}&=-r\sin\theta\sin\phi\ \mathbf{\hat x} + r\sin\theta\cos\phi\ \mathbf{\hat y}

\end{split}

\end{equation}Fair enough so far. I'm simultaneously reading 'A Visual Introduction to Differential Forms and Calculus on Manifolds' by Fortney. In that, a vector ##v_p## in the tangent space ##T_p(\mathbb{R}^3)## is defined as an operator acting on a real function ##f## defined on the manifold (i.e. ##f:\mathbb{R}^3\to\mathbb{R}##). The operator gives the directional derivative of ##f## in the direction ##v_p## at point ##p##:

$$v_p[f]=\sum_{i=1}^3v_i\frac{\partial f}{\partial x^i}\ \bigg|_p$$From this, we can identify the basis vectors of ##T_p(\mathbb{R}^3)## as:

$$\frac{\partial}{\partial x^1}\ \bigg|_p,\frac{\partial}{\partial x^2}\ \bigg|_p,\frac{\partial}{\partial x^3}\ \bigg|_p$$which, as you can see, are quite different from the basis vectors ##\mathbf{e}_u,\mathbf{e}_v,\mathbf{e}_w## that I defined at the start of the question.

**So now let's say I switch to spherical coordinate system and want to specify the tangent space basis vectors in the spherical coordinate representation. And I want to reconcile those basis vectors' spherical representation with the ##\mathbf{e}_r,\mathbf{e}_{\theta},\mathbf{e}_{\phi}## formulas from the Luscombe book that I listed at the beginning.**Now you've seen that the Luscombe book formulas for ##\mathbf{e}_r,\mathbf{e}_{\theta},\mathbf{e}_{\phi}##, that I listed at the start, contain ##\mathbf{r}##. My interpretation of ##\mathbf{r}## is that it's a function from ##\mathbb{R}^3\to\mathbb{R}##, and it's a triple consisting of the three

**coordinate functions**required to specify coordinates of any point ##p## in the manifold ##\mathbb{R}^3##. We can specify ##\mathbf{r}=(x,y,z)##, where\begin{equation}

\begin{split}

x&=x(r,\theta,\phi)&=r\sin\theta\cos\phi \\

y&=y(r,\theta,\phi)&=r\sin\theta\sin\phi \\

z&=z(r,\theta,\phi)&=r\cos\theta

\end{split}

\end{equation}are the individual coordinate functions. Since we've switched to spherical coordinates, we're now using the parameters ##r,\theta,\phi## to specify any point ##p## in the manifold ##\mathbb{R}^3##.

Using the operator definition of tangent vectors,

**if I take ##f=\mathbf{r}=(x,y,z)## and ##v_p=(\mathbf{e}_r)_p\implies v_r=1,v_{\theta}=0,v_{\phi}=0##**, then\begin{equation}

\begin{split}

(\mathbf{e}_r)_p[\mathbf{r}]&=v_r\frac{\partial \mathbf{r}}{\partial r}\ \bigg|_p+

v_{\theta}\frac{\partial \mathbf{r}}{\partial \theta}\ \bigg|_p+

v_{\phi}\frac{\partial \mathbf{r}}{\partial \phi}\ \bigg|_p=\frac{\partial \mathbf{r}}{\partial r}\ \bigg|_p \\

&=\frac{\partial}{\partial r}(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)\ \bigg|_p \\

&=(\sin\theta\cos\phi\,\sin\theta\sin\phi,\cos\theta)\ |_p

\end{split}

\end{equation} which matches the definition mentioned in the Luscombe book. Similarly we can calculate for the ##\theta## and ##\phi## basis vectors.

**Two questions:**1. Does the above procedure of reconciling the two definitions for basis vectors seem correct?

2. If correct, to reconcile the definitions, I had to make a specific assumption about ##f## and act the

*math book version of the basis vector*(as an operator) on ##f=\mathbf{r}## in order to actually get the*physics book version of the same basis vector*. Does that mean the*physics book versions (formulas) of basis vectors*are restrictive and will be incorrect in some scenarios? Or can I just take the physics book versions as the standard definition without worrying too much?If you've read this far, thanks so much for the time and I'd appreciate any help!

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