Objects that have both upper and lower indices, reflecting both types of transformation. 3. The Metric Tensor ( gijg sub i j end-sub
Tensors are defined by how their components transform during a change of coordinates. There are two primary types of transformation: Contravariant ( Aicap A to the i-th power Principles of Tensor Calculus: Tensor Calculus
The fundamental goal of tensor calculus is . While the components of a tensor (like the numbers in a vector) change when you switch from, say, Cartesian to polar coordinates, the physical "object" they represent does not. A tensor equation that is true in one coordinate system is true in all. 2. Transformation Rules Objects that have both upper and lower indices,
Tensor calculus allows us to write "coordinate-free" laws. Instead of writing separate equations for There are two primary types of transformation: Contravariant
It acts as a bridge, allowing you to "lower" a contravariant index to make it covariant, or "raise" it using its inverse ( gijg raised to the i j power
): Components that transform "against" the coordinate change (e.g., position or velocity). They are denoted with upper indices. Covariant ( Aicap A sub i