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docs/en/physics/mathematical-physics/vector-analysis/curvilinear-coordinates.md
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@ -6,7 +6,7 @@ In this section curvilinear coordinate systems will be presented, these are coor
## Covariant basis
> *Definition*: consider a coordinate system $(x_1, x_2, x_3)$ that is defined by the function $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$. Producing a position vector for every combination of coordinate values.
> *Definition*: consider a coordinate system $(x_1, x_2, x_3)$ that is mapped by $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ with respect to a reference coordinate system. Producing a position vector for every combination of coordinate values.
>
> * For two coordinates fixed, a coordinate curve is obtained.
> * For one coordinate fixed, a coordinate surface is obtained.
@ -21,7 +21,7 @@ We will now use this coordinate system described as $\mathbf{x}$ to formulate a
>
> for all $(x_1, x_2, x_3) \in \mathbb{R}^3$ and $i \in \{1, 2, 3\}$.
Obtaining basis vectors that are tangential to the corresponding coordinate curves. Therefore any vector $\mathbf{u} \in \mathbb{3}$ can be written in terms of its components with respect to this basis
Obtaining basis vectors that are tangential to the corresponding coordinate curves. Therefore any vector $\mathbf{u} \in \mathbb{R}^3$ can be written in terms of its components with respect to this basis
$$
\mathbf{u} = \sum_{i=1}^3 u_i \mathbf{a}_i
@ -99,7 +99,7 @@ with $h_i = \sqrt{\langle \mathbf{a}_i, \mathbf{a}_i \rangle} = \|\mathbf{a}_i\|
The covariant basis vectors have been constructed as tangential vectors of the coordinate curves. An alternative basis can be constructed from vectors that are perpendicular to coordinate surfaces.
> *Definition*: for a valid set of covariant basis vectors the contravariant basis vectors may be defined given by
> *Definition*: for a valid set of covariant basis vectors the contravariant basis vectors may be defined, given by
>
> $$
> \begin{align*}
@ -117,22 +117,23 @@ From this definition it follows that $\langle \mathbf{a}^i, \mathbf{a}_j \rangle
> \delta_{ij} = \begin{cases} 1 &\text{ if } i = j, \\ 0 &\text{ if } i \neq j.\end{cases}
> $$
Also a metric tensor for contravariant basis vectors can be defined with it the relations between covariant and contravariant quantities can be found.
A metric tensor for contravariant basis vectors may be defined. With which the relations between covariant and contravariant quantities can be found.
> *Definition*: the components of the metric tensor for contravariant basis vectors are defined as
>
> $$
> g^{ij} := \langle \mathbf{a}^i, \mathbf{a}^j \rangle,
> $$
>
> therefore the metric tensor for contravariant basis vectors is given by
>
> $$
> (g^{ij}) = \begin{pmatrix} \langle \mathbf{a}^1, \mathbf{a}^1 \rangle & \langle \mathbf{a}^1, \mathbf{a}^2 \rangle & \langle \mathbf{a}^1, \mathbf{a}^3 \rangle \\ \langle \mathbf{a}^2, \mathbf{a}^1 \rangle & \langle \mathbf{a}^2, \mathbf{a}^2 \rangle & \langle \mathbf{a}^2, \mathbf{a}^3 \rangle \\ \langle \mathbf{a}^3, \mathbf{a}^1 \rangle & \langle \mathbf{a}^3, \mathbf{a}^2 \rangle & \langle \mathbf{a}^3, \mathbf{a}^3 \rangle \end{pmatrix}.
> $$
<br>
These relations are stated in the proposition below.
> *Lemma*: considering the two ways of representing the vector $\mathbf{u} \in \mathbb{R}^3$ given by
> *Proposition*: considering the two ways of representing the vector $\mathbf{u} \in \mathbb{R}^3$ given by
>
> $$
> \mathbf{u} = u^i \mathbf{a}_i = u_i \mathbf{a}^i.
@ -244,7 +245,7 @@ $$
\|\mathbf{u}\| = \sqrt{u^{(i)} u_{(i)}}.
$$
We will discuss as an example the representations of the cartesian, cylindrical and spherical coordinate systems viewed from a cartesian perspective. This means that the coordinate maps are based on the cartesian interpretation of then. Every other interpretation could have been used, but our brains have a preference for cartesian it seems.
We will discuss as an example the representations of the cartesian, cylindrical and spherical coordinate systems viewed from a cartesian perspective. This means that the coordinate maps are based on the cartesian interpretation of then. Every other interpretation could have been used, but our brains have a preference for cartesian it seems.z
Let $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ map a cartesian coordinate system given by