same idea can be expressed in matrix and index notations (including the summation convention), respectively: By "linear", if for two scalars ρ and σ and vectors r and s, then in function and index notations: and similarly for the matrix notation. The function, matrix, and index notations all mean the same thing. The matrix forms provide a clear display of the components, while the index form allows easier tensor-algebraic manipulation of the formulae in a compact manner. Both provide the physical interpretation of directions; vectors have one direction, while second-order tensors connect two directions together. One can associate a tensor index or coordinate label with a basis vector direction. The use of second-order tensors are the minimum to describe changes in magnitudes and directions of vectors, as the dot product of two vectors is always a scalar, while the cross product of two vectors is always a pseudovector perpendicular to the plane defined by the vectors, so these products of vectors alone cannot obtain a new vector of any magnitude in any direction. (See also below for more on the dot and cross products). The tensor product of two vectors is a second-order tensor, although this has no obvious directional interpretation by itself. The previous idea can be continued: if T takes in two vectors p and q, it will return a scalar r. In function notation we write r = T(p, q), while in matrix and index notations (including the summation convention) respectively: The tensor T is linear in both input vectors. When vectors and tensors are written without reference to components, and indices are not used, sometimes a dot ⋅ is placed where summations over indices (known as tensor contractions) are taken. For the above cases: motivated by the dot product notation: More generally, a tensor of order m which takes in n vectors (where n is between 0 and m inclusive) will return a tensor of order , see for further generalizations and details. The concepts above also apply to pseudovectors in the same way as for vectors. The vectors and tensors themselves can vary within throughout space, in which case we have vector fields and tensor fields, and can also depend on time. Following are some examples: {| class="wikitable" |- ! scope="col" | An applied or given... ! scope="col" | ...to a material or object of... ! scope="col" | ...results in... ! scope="col" | ...in the