The derivation proves that the partial derivatives of a scalar field, $\partial_a \phi$, naturally form the covariant components of a vector. This is a fundamental concept in tensor calculus because a scalar field's value is independent of the coordinate system. By applying the chain rule to a coordinate transformation, the partial derivatives are shown to transform in a manner identical to the definition of a covariant vector. This means the transformation rule for a covariant vector, $V_b^{\prime}= \sum_a \frac{\partial x^a}{\partial x^b} V_a$, perfectly matches the transformation of the partial derivatives, $\frac{\partial \phi^{\prime}}{\partial x^b}=\sum_a \frac{\partial \phi}{\partial x^a} \frac{\partial x^a}{\partial x^b}$. This result validates that the gradient, which is a vector composed of these partial derivatives, is a quintessential example of a covariant vector.
A derivative illustration based on our specific text and creative direction
A derivative illustration based on our specific text and creative direction
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
‣
<aside> <img src="/icons/report_pink.svg" alt="/icons/report_pink.svg" width="40px" />
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]
©️2026 Sayako Dean
</aside>