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Wikipedia : Tensor Calculus
By definition, the scalars of a vector space are its tensors of rank 0.
In any vector space, a linear function which sends a vector to a scalar may be called a covector. Normally, covectors and vectors are different types of things. (Think of the bras and kets of quantum mechanics.) However, if we are considering only finitely many dimensions, then the space of vectors and the space of covectors have the same number of dimensions and can therefore be put in a linear one-to-one correspondence with each other.
Such a bijective correspondence is called a metric and is fully specified by a nondegenerate quadratic form, denoted by a dot-product ("nondegenerate" precisely means that the associated correspondence is bijective).
A metric is said to be Euclidean if it is "positive definite", which is to say that V.V is positive for any nonzero vector V. Euclidean metrics are nondegenerate but other metrics exist which are nondegenerate in the above sense without being "definite" (which is to say that V.V can be zero even when V is nonzero). Such metrics are perfectly acceptable. They include the so-called Lorentzian metric of four-dimensional spacetime, which is our primary concern here.
Once a metric is defined, we are allowed to blur completely the distinction between vectors and covectors as they are now in canonical one-to-one correspondence. A tensor of rank zero is a scalar.
More generally, a tensor of nonzero rank n (also called nth-rank tensor, or n-tensor) is a linear function that maps a vector to a tensor of rank n-1.
Such an object is intrinsically defined, although it can be specified by either its covariant or its contravariant coordinates in a given basis (cf. 2D example).
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Tensor Analysis by Pavel Grinfeld (video 2): The Two Definitions of the Gradient
