But they are not totally different! The Physicists' tensors are actually Mathematicians' tensors, but parametrized by some parameters (coordinates). Then you ha

But they are not totally different! The Physicists' tensors are actually Mathematicians' tensors, but parametrized by some parameters (coordinates). Then you have special laws what happens if you make a (possibly non-linear) change of the coordinates. See https://en.wikipedia.org/wiki/Tensor#Tensor_fields https://en.wikipedia.org/wiki/Tensor#Tensor_fields