Is a tensor just a specific type of matrix that we use often? or does it have distinct behavior? How do we understand the idea of Quotient rule in tensors?
The Quotient rule is essentially a theorem that lets you determine if some mathematical object labeled with indices is a truly Tensor or not. For example, just because some mathematical object is labeled as T_{jkl} does not automatically mean that T_{jkl} is a rank-3 Tensor. Whether some object is a Tensor or not depends on whether it transforms under rotations within a coordinate system leaving some of its key properties invariant. For example, the length of a vector (rank-1 Tensor) does not change when the vector is rotated – therefore vectors are Tensors. Higher rank Tensors have additional properties that remain invariant, in analogy to the length of a vector, under rotations. It is the invariance of those properties under rotations that proves that the object is indeed a Tensor. Back to the Quotient rule: it basically states the following – (1) take linear combinations of Tensor elements with elements of an input object that you think maybe a Tensor. (2) If you can show that the resulting output object is a Tensor, then the input object is also a Tensor.