🧄Derivation of Tensor Transformation Properties for Mixed Tensors

A mixed tensor transforms by applying the Jacobian to its contravariant components and the inverse Jacobian to its covariant components, demonstrating that each index transforms independently based on its type.

For any physical vector, its components can be described in two distinct ways when the coordinate system is not orthogonal. Contravariant Components tell you how to build or get to the tip of the vector by following the grid lines. In the demo, the orange and green arrows in this mode show you must travel a certain distance parallel to the first axis, and then a certain distance parallel to the second axis to arrive at the destination. They are the coordinates you would use to give someone directions on your skewed map. Covariant Components tell you how to measure the vector's projection onto each axis. Imagine shining a flashlight from a great distance, perpendicular to an axis. The length of the vector's shadow on that axis is the covariant component. It's a direct measurement of "how much" of the vector points along that specific direction.

Derivation of Tensor Transformation Properties for Mixed Tensors | Proof and Derivationviadean on Notion