ayey (a vector), and similarly for x and z. A more general notation is tensor index notation, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are replaced by tensor indices in the basis vectors ex ↦ e1, ey ↦ e2, ez ↦ e3 and coordinates ax ↦ a1, ay ↦ a2, az ↦ a3. In general, the notation e1, e2, e3 refers to any basis, and a1, a2, a3 refers to the corresponding coordinate system; although here they are restricted to the Cartesian system. Then: It is standard to use the Einstein notation—the summation sign for summation over an index that is present exactly twice within a term may be suppressed for notational conciseness: An advantage of the index notation over coordinate-specific notations is the independence of the dimension of the underlying vector space, i.e. the same expression on the right hand side takes the same form in higher dimensions (see below). Previously, the Cartesian labels x, y, z were just labels and not indices. (It is informal to say "i = x, y, z"). Second-order tensors in three dimensions A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b. Analogous to vectors, it can be written as a linear combination of the tensor basis , , ..., (the right-hand side of each identity is only an abbreviation, nothing more): Representing each basis tensor as a matrix: then T can be represented more systematically as a matrix: See matrix multiplication for the notational correspondence between matrices and the dot and tensor products. More generally, whether or not T is a tensor product of two vectors, it is always a linear combination of the basis tensors with coordinates Txx, Txy, ..., Tzz: while in terms of tensor indices: and in matrix form: Second-order tensors occur naturally in physics and engineering when physical quantities have directional dependence in the system, often in a "stimulus-response" way. This can be mathematically seen through one aspect of tensors – they are multilinear functions. A second-order tensor T which takes in a vector u of some magnitude and direction will return a vector v; of a different magnitude and in a different direction to u, in general. The notation used for functions in mathematical analysis leads us to write , while the