Calculus of Partial Differentials

In a two-dimensional domain (x, y) the partial differential of a function g(x, y) in each direction is given as 

A geometrical interpretation of the above is shown in Fig.B.l. For a function h(x,y,z) in a three-dimensional domain (x,y,z) we can have similar partial differentials. 

be an orthogonal basis in The gradient of a scalar-valued function 

Ichikawa and A.P.S. Selvadurai, Transport Phenomena in Porous Media, DOI 10.1007/978-3-642-25333-1, Springer-Verlag Berlin Heidelberg 2012 

Let h x) = hiCi be a vector-valued function, e.g., a displacement u x). The gradient of h x) is given as a second order tensor, and the components are written in a matrix form as 

---