In non-orthogonal coordinate systems, a vector is physically constructed from a tangent basis but measured through a dual basis, establishing a reciprocal relationship where the dual vectors function as directional filters. This relationship allows contravariant components ($v^a$) to be extracted via the dot product $\vec{E}^a \cdot \vec{v}$, mirroring the method used to find covariant components ($v_a$) with the tangent basis. Central to this process is the orthogonality condition ($\vec{E}^a \cdot \vec{E}_b = \delta_b^a$), which ensures that a dual vector can "sift out" a specific component by ignoring contributions from other basis vectors,. Crucially, the dual basis dynamically compensates for geometric shifts; if the tangent vectors collapse toward one another, the dual vectors stretch and rotate outward to maintain the mathematical integrity required to recover the vector's fixed components.
A derivative illustration based on our specific text and creative direction
A derivative illustration based on our specific text and creative direction
The sequence diagram and state diagram serve as functional and behavioral maps for the derivation sheet, translating its abstract proofs into a clear timeline of events and a set of real-world scenarios. While the derivation sheet provides the formal rules, these diagrams explain the operational logic and the geometric consequences of those rules without the need for complex formulas.
block-beta
columns 6
CC["Criss-Cross"]:6
%% Condensed Notes
CN["Condensed Notes"]:6
RF["Relevant File"]:6
NV["Narrated Video"]:6
PA("Plotting & Analysis")AA("Animation & Analysis")KT("Summary & Interpretation") ID("Illustration & Demo") VA1("Visual Aid")MG1("Multigraph")
%% Proof and Derivation
PD["Proof and Derivation"]:6
AF("Derivation Sheet"):6
NV2["Narrated Video"]:6
PA2("Plotting & Analysis")AA2("Animation & Analysis")KT2("Summary & Interpretation") ID2("Illustration & Demo")VA2("Visual Aid") MG2("Multigraph")
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
class VA1 color_3
classDef color_4 fill:#47a291,stroke:#47a291,color:#47a291
class VO color_4
class MG1 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
class VA2 color_6
classDef color_7 fill:#47a291,stroke:#47a291,color:#fff
class VO2 color_7
class MG2 color_7
‣