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Partial derivative

While most of the expressions of the Ricci calculus are valid for arbitrary bases, the expressions involving partial derivatives of tensor components with respect to coordinates apply only with a coordinate basis: a basis that is defined through differentiation with respect to the coordinates. Coordinates are typically denoted by x, but do not in general form the components of a vector. In flat spacetime with linear coordinatization, a tuple of differences in coordinates, Δ x, can be treated as a contravariant vector. With the same constraints on the space and on the choice of coordinate system, the partial derivatives with respect to the coordinates yield a result that is effectively covariant. Aside from use in this special case, the partial derivatives of components of tensors do not in general transform covariantly, but are useful in building expressions that are covariant, albeit still with a coordinate basis if the partial derivatives are explicitly used, as with the covariant, exterior and Lie derivatives below.

To indicate partial differentiation of the components of a tensor field with respect to a coordinate variable x, a comma is placed before an appended lower index of the coordinate variable.

This may be repeated (without adding further commas):

These components do not transform covariantly, unless the expression being differentiated is a scalar. This derivative is characterized by the product rule and the derivatives of the coordinates

where δ is the Kronecker delta.