a general vector space , the space is sometimes referred to as a coordinate space. Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space assumes a choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space. Computationally, it is necessary to work with the more concrete description in order to do concrete computations. Affine description A more abstract description still is to model physical space as a three-dimensional affine space over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of , the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces. This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance. Inner product space The above discussion does not involve the dot product. The dot product is an example of an inner product. Physical space can be modelled as a vector space which additionally has the structure of an inner product. The inner product defines notions of length and angle (and therefore in particular the notion of orthogonality). For any inner product, there exist bases under which the inner product agrees with the dot product, but again, there are many different possible bases, none of which are preferred. They differ from one another by a rotation, an element of the group of rotations SO(3). In calculus Gradient, divergence and curl In a rectangular coordinate system, the gradient of a (differentiable) function is given by and in index notation is written The divergence of a (differentiable) vector field F = U i + V j + W k, that is, a function , is equal to the scalar-valued function: In index notation, with Einstein summation convention this is Expanded