Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mechanics.

Vector and tensor algebra in three-dimensional curvilinear coordinates

Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid 1900s, for example the text by Green and Zerna.^[1] Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,^[2] Naghdi,^[3] Simmonds,^[4] Green and Zerna,^[1] Basar and Weichert,^[5] and Ciarlet.^[6]

Coordinate transformations

Consider two coordinate systems with coordinate variables ${\displaystyle (Z^{1},Z^{2},Z^{3})}$ and ${\displaystyle (Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})}$, which we shall represent in short as just ${\displaystyle Z^{i}}$ and ${\displaystyle Z^{\acute {i}}}$ respectively and always assume our index ${\displaystyle i}$ runs from 1 through 3. We shall assume that these coordinates systems are embedded in the three-dimensional euclidean space. Coordinates ${\displaystyle Z^{i}}$ and ${\displaystyle Z^{\acute {i}}}$ may be used to explain each other, because as we move along the coordinate line in one coordinate system we can use the other to describe our position. In this way Coordinates ${\displaystyle Z^{i}}$ and ${\displaystyle Z^{\acute {i}}}$ are functions of each other

$${\displaystyle Z^{i}=f^{i}(Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})}$$

${\displaystyle i=1,2,3}$

which can be written as