Adding Covariant/Contravariant Vector Documentation and Visualization

docs/refs.bib

@@ -271,3 +271,11 @@ @article{Wiin1967
   year = {1967},
   publisher = {Wiley Online Library}
 }
+
+@article{Yatunin2025,
+  title = {The Climate Modeling Alliance Atmosphere Dynamical Core: Concepts, Numerics, and Scaling},
+  author = {Dennis Yatunin, Simon Byrne, Charles Kawczynski, Sriharsha Kandala, Gabriele Bozzola, Akshay Sridhar, Zhaoyi Shen, Anna Jaruga, Julia Sloan, Jia He, Daniel Zhengyu Huang, Valeria Barra, Oswald Knoth, Paul Ullrich, Tapio Schneider},
+  doi = {10.22541/essoar.173940262.23304403/v1},
+  year = {2025},
+  publisher = {ESS Open Archive}
+}

docs/src/math_framework.md

@@ -62,6 +62,19 @@ then may want to consider different bases, as not all operators accept all bases
 _Covariance_ and _contravariance_ describe how the quantitative description of
 certain geometric or physical entities changes with a change of basis.
 
+More specifically, 
+
+__Covariant objects__ —whether you mean covariant components (the coefficients that sit in front of the basis) 
+or the covariant basis vectors themselves—co-vary with the coordinate grid. In other words, when you change coordinates, 
+these quantities transform in the same way as the coordinate differentials. By convention they carry lower indices (subscripts).
+
+__Contravariant objects__—whether you mean contravariant components or the contravariant basis vectors—vary contra 
+to the coordinate grid. That is, they transform in the opposite way to the coordinate differentials so as to keep 
+tensorial expressions invariant. By convention they carry upper indices (superscripts).
+
+In ClimaCore.jl, `CovariantVector`s are aligned with the _contravariant basis vectors_, but have _covariant components_. 
+Conversely, `ContravariantVector`s are aligned with the _covariant basis vectors_, but have _contravariant components_. 
+
 In ClimaCore.jl, the _covariant basis_ is specified by the partial derivative
 of the transformation from the reference element ``\xi \in [-1,1]^d`` (where ``d``
 is the dimensionality of the domain ``\Omega``) to ``x`` in the physical space:
@@ -74,6 +87,48 @@ while the _contravariant basis_ is the opposite: gradient in ``x`` of the coordi
 \mathbf{e}^i = \nabla_x \xi^i
 ```
 
+If we plot these bases in a curvilinear space, _covariant basis_ vectors “ride along” the coordinate surface liness (parallel),  while _contravariant basis_ vectors “stick out” of those surface lines (perpendicular). See the plot from [Yatunin2025](@cite) below:
+
+![normal and tangent](normal_tangent.png)
+
+Here is a visual representation of how vectors can be represented in _contravariant_ and _covariant_ components.
+
+![Different bases](contrava_cova.png)
+
+From the above two figures, we can see that parallel projections would lead to contravariant components $a^{1}$ and $a^{2}$, 
+while perpendicular projections would lead to covariant components $b_{1} = \vec{b}\cdot \vec{e_{1}}$ and $b_{2} = \vec{b}\cdot \vec{e_{2}}$.
+
+As to better connect the original idea of _covariant components_ / _contravariant components_ with the real application in ClimaCore.jl, 
+we bring the case of __polar coordinates__ -- a classic example of a __curvilinear coordinate system__.
+
+First of all, we have the Polar‐to‐Cartesian mapping as following:
+```math
+\vec{r} (r,\theta ) = (r \cos \theta, r \sin \theta)
+```
+
+Then we have the _covariant basis_:
+```math
+\vec{e_{r} } = \frac{\partial \vec{r} }{\partial r}  = \frac{\partial (r \cos \theta, r \sin \theta) }{\partial r} = (\cos \theta, \sin \theta)
+```
+```math
+\vec{e_{\theta} } = \frac{\partial \vec{r} }{\partial \theta }  = \frac{\partial (r \cos \theta, r \sin \theta) }{\partial \theta } = (-r\sin \theta,r \cos \theta)
+```
+
+$\vec{e_{r} }$ represents the direction that is tangent to the "radius direction", and $\vec{e_{\theta} }$ is tangent to the circle curve itself.
+
+And the _contravariant basis_:
+
+since we have $r(x,y) = \sqrt{x^{2} +y^{2}  }$ and $\theta (x,y) = \arctan (\frac{y}{x} )$. Thus,
+
+```math
+\vec{e^{r} } = \nabla r(x,y) = (\frac{x}{\sqrt{x^{2} +y^{2}  }}, \frac{y}{\sqrt{x^{2} +y^{2}  }}) = (\frac{x}{r},\frac{y}{r} ) = (\cos \theta, \sin \theta)
+```
+```math
+\vec{e^{\theta } } = \nabla \theta(x,y) = (\frac{-y}{x^{2} +y^{2}}, \frac{x}{x^{2} +y^{2}})  = (\frac{-\sin \theta}{r} , \frac{\cos \theta}{r})
+```
+
+$\vec{e^{r} }$ represents the direction that is perpendicular to the circle curve, and $\vec{e^{\theta} }$ is perpendicular to the "radius direction".
+
 
 **Note**:
 
@@ -85,6 +140,9 @@ while the _contravariant basis_ is the opposite: gradient in ``x`` of the coordi
 
 * things get a little more complicated in the presence of terrain, but ``\xi^3`` is radially aligned
   - the 3rd covariant component is aligned with W, but the 3rd contravariant component may not be (e.g. at the surface it is normal to the boundary).
+
+
+
 
 ### Cartesian bases
 Analogously to `CartesianPoint`s, in ClimaCore, there are also `CartesianVector`s: