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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 ## 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 two coordinates fixed, a coordinate curve is obtained.
> * For one coordinate fixed, a coordinate surface 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 $i \in \{1, 2, 3\}$. > 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 \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. 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*} > \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} > \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 > *Definition*: the components of the metric tensor for contravariant basis vectors are defined as
> >
> $$ > $$
> g^{ij} := \langle \mathbf{a}^i, \mathbf{a}^j \rangle, > g^{ij} := \langle \mathbf{a}^i, \mathbf{a}^j \rangle,
> $$ > $$
>
> therefore the metric tensor for contravariant basis vectors is given by > 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}. > (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. > \mathbf{u} = u^i \mathbf{a}_i = u_i \mathbf{a}^i.
@ -244,7 +245,7 @@ $$
\|\mathbf{u}\| = \sqrt{u^{(i)} u_{(i)}}. \|\mathbf{u}\| = \sqrt{u^{(i)} u_{(i)}}.
$$ $$
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 Let $\mathbf{x}: \mathbb{R}^3 \to \mathbb{R}^3$ map a cartesian coordinate system given by