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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In physics, a covariant transformation is a rule (specified below), that describes how certain physical entities change under a change of coordinate system. In particular the term is used for vectors and tensors. The transformation that describes the new basis vectors in terms of the old basis, is defined as a covariant transformation. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities that transform in the same way. The inverse of the covariant transformation is called the contravariant transformation. In order that a vector should be invariant under a coordinate transformation, the components of a vector must transform according to the contravariant rule. Conventionally, indices identifying the components of a vector are placed as upper indices and so are all indices of entities that transform in the same way. The summation over all indices of a product with the same lower and upper indices are invariant to a transformation.