advanced-linear-algebra

Covariant, contravariant, and mixed tensors

taught

A tensor built from copies of V (contravariant) transforms differently under a change of basis than one built from copies of $V^{*}$ (covariant) -- the distinction chapter 03's duality already set up.

Taught in

Covariant, contravariant, and mixed tensors
Advanced Linear Algebra

Requires

bases-and-coordinates-for-tensors required
the-dual-space required

Required by

Contraction and the trace required

Local knowledge map

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Solid arrows are required prerequisites; dashed are recommended or helpful. Lower rows are more foundational. Click any box to jump to that concept.