Solving for Metric Tensors and Christoffel Symbols

The hyperbolic and parabolic coordinate systems, though both curvilinear, reveal different geometric properties through their metric tensors. For the hyperbolic system, the non-diagonal metric tensor indicates that the coordinate lines are not orthogonal, meaning the basis vectors at any given point are not perpendicular. The metric components, and consequently the scale factors, vary with both coordinates, highlighting a non-uniform and non-flat geometry. In contrast, the parabolic coordinate system is orthogonal, as shown by its diagonal metric tensor. While the basis vectors are perpendicular, their magnitudes (the scale factors) still depend on the coordinates. This change in scale means that while the coordinate grid is "square" in a generalized sense, the distance represented by a unit change in a coordinate varies with location. In both systems, the non-zero Christoffel symbols are a natural consequence of the changing basis vectors, which is a fundamental characteristic of curvilinear coordinates.

🎬 Narrated Video

https://youtube.com/shorts/1jvmRtzxQHo?feature=share

📎IllustraDemo

A derivative illustration based on our specific text and creative direction

🏗️Structural clarification of Poof and Derivation

block-beta
columns 5
CC["Criss-Cross"]:5

%% Condensed Notes

CN["Condensed Notes"]:5
RF["Relevant File"]:5
NV["Narrated Video"]:4 VO["Voice-over"] 
PA("Plotting & Analysis")AA("Animation & Analysis")KT("Summary & Interpretation") ID("Illustration & Demo") PO("Polyptych")

%% Proof and Derivation

PD["Proof and Derivation"]:5
AF("Derivation Sheet"):5
NV2["Narrated Video"]:4 VO2["Voice-over"]
PA2("Plotting & Analysis")AA2("Animation & Analysis")KT2("Summary & Interpretation") ID2("Illustration & Demo") PO2("Polyptych")

classDef color_1 fill:#8e562f,stroke:#8e562f,color:#fff
class CC color_1

%% %% Condensed Notes

classDef color_2 fill:#14626e,stroke:#14626e,color:#14626e
class CN color_2
class RF color_2

classDef color_3 fill:#1e81b0,stroke:#1e81b0,color:#1e81b0
class NV color_3
class PA color_3
class AA color_3
class KT color_3
class ID color_3

classDef color_4 fill:#47a291,stroke:#47a291,color:#47a291
class VO color_4
class PO color_4

%% Proof and Derivation

classDef color_5 fill:#307834,stroke:#307834,color:#fff
class PD color_5
class AF color_5

classDef color_6 fill:#38b01e,stroke:#38b01e,color:#fff
class NV2 color_6
class PA2 color_6
class AA2 color_6
class KT2 color_6
class ID2 color_6

classDef color_7 fill:#47a291,stroke:#47a291,color:#fff
class VO2 color_7
class PO2 color_7

🗒️Downloadable Files - Recursive updates

‣

Copyright Notice

All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

</aside>