42
2 Vector and Tensor Analysis
since the base vectors are constant. For polar coordinates, the derivatives of e r and
e θ were determined in Example 2.1. Using Eqs. (2.3) gives
u, r = u r , r e r + u r e r , r +u θ , r e θ + u θ e θ , r
= u r , r e r + u θ , r e θ
u, θ = u r , θ e r + u r e r , θ +u θ , θ e θ + u θ e θ , θ
= (u r , θ −u θ )e r + (u r + u θ , θ )e θ .
2.6.2 Differentiation with Respect to Vectors and Tensors
Equations written in direct notation sometimes contain derivatives of functions
taken with respect to vectors and tensors. Here, we use the chain rule to define
the total differential of the scalar function φ(a, T) in the form
dφ = da ·
∂φ
∂a
+ dT :
∂φ
∂T
.
(2.56)
In terms of components, the vector and tensor differentials in this equation can be
written
da = da i e i
dT = dT ij e i e j ,
(2.57)
which are used to define the operators
∂
∂a
= e i
∂
∂a i
∂
∂T
= e i e j
∂
∂T ij
.
(2.58)
These expressions can be confirmed by inserting them into Eq. (2.56) to get
dφ = da ·
∂φ
∂a
+ dT :
∂φ
∂T
= (da i e i ) ·
e j
∂φ
∂a j
+ (dT ij e i e j ) :
e k e l
∂φ
∂T kl