orientation of the coordinate system. The totally anti-symmetric symbol nevertheless allows a convenient handling of the cross product in equally oriented three dimensional coordinate systems. This table shows important examples of tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type , where n is the number of contravariant indices, m is the number of covariant indices, and gives the total order of the tensor. For example, a bilinear form is the same thing as a -tensor; an inner product is an example of a -tensor, but not all -tensors are inner products. In the -entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor. Raising an index on an -tensor produces an -tensor; this corresponds to moving diagonally down and to the left on the table. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. Contraction of an upper with a lower index of an -tensor produces an -tensor; this corresponds to moving diagonally up and to the left on the table. Properties Assuming a basis of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized multidimensional array of numerical values with respect to this specific basis. Changing the basis transforms the values in the array in a characteristic way that allows to define tensors as objects adhering to this transformational behavior. For example, there are invariants of tensors that must be preserved under any change of the basis, thereby making only certain multidimensional arrays of numbers a tensor. Compare this to the array representing not being a tensor, for the sign change under transformations changing the orientation. Because the components of vectors and their duals transform differently under the change of their dual bases, there is a covariant and/or contravariant transformation law that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. The numbers of, respectively, (contravariant indices) and dual (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers , which determine the precise form of the transformation law. The