I don’t know why the definition of tensor is so mysterious: tensor is something transform like a tensor. Sho Takatori also pointed out the absurdity and illuminate it with Einstein index notation in real calculus computation. Nevertheless, it’s still quite complicated, that’s why genius Einstein had to come in and provide a notation for generations afterwards.

He is the first one clearly illustrate what gradient X versus gradient dot X is: the former one is a matrix and latter is a scalar.

further illustrate the power by