The derivation of the tensor transformation properties for mixed tensors establishes the rule that a mixed tensor of type (n, m), having n contravariant (upper) indices and m covariant (lower) indices, transforms according to the independent action of each index type. The core principle is that a scalar formed by contracting the mixed tensor with appropriate covariant and contravariant vectors must be invariant under coordinate changes. This leads to the transformation rule: each contravariant index transforms with a factor of the Jacobian $\left(\frac{\partial y^{a'}}{\partial y^a}\right)$, and each covariant index transforms with a factor of the inverse Jacobian $\left(\frac{\partial y^b}{\partial y^{b'}}\right)$. The overall transformation is the product of n Jacobian factors and m inverse Jacobian factors.
block-beta
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CC["Criss-Cross"]:5
%% Condensed Notes
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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
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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")
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%% %% Condensed Notes
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%% Proof and Derivation
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