Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 389

This operator is called a vector operator because it increases the tensorial order
of the quantity operated upon by one. For example, the gradient of a scalar function
f ( x 1 , x 2 , x 3 , t ) is a vector given by

Þ¼ @

x 1 e 1 þ @

f

x 2 e 2 þ @

f f

r f ð x 1 ; x 2 ; x 3 ; t

x 3 e 3 :

(A.172)

@ @ @

To verify that the gradient operator transforms as a vector consider the operator
in both the Latin and Greek coordinate systems,

Þ¼ @

f

Þ¼ @

f

r ð L Þ f x ð L Þ ; r ð G Þ f x ð G Þ ;

ð t

x i e i and

ð t e a ; (A.173)

@ @ x a

respectively, and note that, by the chain rule of partial differentiation,

@

x i ¼ @

f f @ x a

x i :

(A.174)

@ @ x a @

Now since, from (A.77), x ð G Þ ¼ Q T x ð L Þ , or index notation x a ¼

Q i a x i it follows

and, from (A.174), @

f

Q i a @

f

that Q i a ¼ @ x a

x i ¼

or

@x i

@ @ x a

r ð L Þ f x ð L Þ ; r ð G Þ f x ð G Þ ;

ð t Þ¼ Q ð t Þ: (A.175)

This shows that the gradient is a vector operator because it transforms like a
vector under changes of coordinate systems.
The gradient of a vector function r ( x 1 , x 2 , x 3 , t ) is a second order tensor given by

Þ¼ @

r j

r r ð x 1 ; x 2 ; x 3 ; t

x i e i

e j ; (A.176)

@

where

2
4

3
5

@ r 1 @ r 1 @ r 1

@ x 1 @ x 2 @ x 3

¼

@ r i

@ r 2 @ r 2 @ r 2

T

½r r ¼ : (A.177)

@ x j @ x 1 @ x 2 @ x 3

@ r 3 @ r 3 @ r 3

@ x 1 @ x 2 @ x 3

As this example suggests, when the gradient operator is applied, the tensorial order
of the quantity operated upon it increases by one. The matrix that is the open product
of the gradient and r is arranged in (A.177) so that the derivative is in the first (or row)
position and the vector r is in the second (or column) position. The divergence
operator is a combination of the gradient operator and a contraction operation that
results in the reduction of the order of the quantity operated upon to one lower than it