# 3/3-Unlocking the Secrets of Shape and Space: A Glimpse into Differential Geometry & Manifolds

> Tangent vectors can be understood in multiple ways, including as equivalence classes of curves, derivations on smooth functions, or velocities of curves, all of which are fundamental to understanding vector fields and other complex concepts like the Lie bracket and torsion tensor in differential geometry.

[This cloud computing explores various concepts within differential geometry, including equivalence classes of curves, derivations on smooth functions, vector fields on 2D and 3D spheres, tangent vectors, and torsion tensors, through animated results, plotting, and computations.](https://viadean.notion.site/3-3-Unlocking-the-Secrets-of-Shape-and-Space-A-Glimpse-into-Differential-Geometry-Manifolds-1fa1ae7b9a32800a9ef0d895a322b57c?source=copy_link)

### 🎬Animated result

{% embed url="<https://youtu.be/wzv_H3-6OHc>" %}
Equivalence Classes of Curves
{% endembed %}

{% embed url="<https://youtu.be/pZDbYTWtBp8>" %}
Derivations on Smooth Functions
{% endembed %}

{% embed url="<https://youtu.be/PbMh0GSLbfI>" %}
the flow of points along the vector field lines on an equirectangular projection
{% endembed %}

{% embed url="<https://youtu.be/Jm_11ctr_9I>" %}
Vector Field on the 3D Sphere
{% endembed %}

{% embed url="<https://youtu.be/qXraQqaD0ts>" %}
Tangent Vectors as Velocities of Curves
{% endembed %}

{% embed url="<https://youtu.be/uTR4SI_RSnw>" %}
Directional Derivative via Curves
{% endembed %}

{% embed url="<https://youtu.be/bOcw9kIk5b4>" %}
Directional Derivative in Coordinates
{% endembed %}

{% embed url="<https://youtu.be/JZZv3VtwtAQ>" %}
Lie Bracket of Vector Fields
{% endembed %}