When the curvilinear coordinate system is generated the transformation
and therefore the set of mapping functions 
, 
is known. Then we are able to derive expressions for the differential
operators in the curvilinear coordinates using the elements of the Jacobian
of the transformation .
Partial derivatives with respect to cartesian coordinates 
are
related to partial derivatives with respect to curvilinear coordinates by
the chain rule which may be written in either of two ways. To start,
we consider a continuously differentiable scalar-valued function 
. Then, we
may use (3.3-44) or, equivalently, (3.3-45) to relate the
cartesian and curvilinear derivatives of the function .
We easily identify in the first case the contravariant base vectors 
and 
, whereas in the second case the covariant base vectors
and 
.
