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2.1 Differential operators

Before defining the spatial derivatives we first introduce the nabla vector operator  \~/  \special {t4ht=(from where Nabla Ltd. takes its name), represented in index notation as @i  \special {t4ht=:

            @    (  @    @    @  )
 \~/   =_  @i  =_ ---- =_   ----,----,----
           @xi     @x1  @x2  @x3
\special {t4ht=
(2.1)

The nabla operator is a useful notation that obeys the following rules:

  • it operates on the tensors to its right and the conventional rules of a derivative of a product, e.g. @iab = (@ia) b + a(@ib)  \special {t4ht=;
  • otherwise the nabla operator behaves like any other vector in an algebraic operation.

2.1.1 Gradient

If a scalar field s  \special {t4ht= is defined and continuously differentiable then the gradient of s  \special {t4ht=,  \~/ s  \special {t4ht= is a vector field

            (               )
 \~/ s = @ s =   -@s-,-@s-,-@s-
        i     @x1  @x2  @x3
\special {t4ht=
(2.2)

The gradient can operate on any tensor field to produce a tensor field that is one rank higher. For example, the gradient of a vector field a  \special {t4ht= is a second rank tensor field

             (                              )
                @a1/@x1   @a2/@x1  @a3/@x1
 \~/ a = @iaj =    @a1/@x2   @a2/@x2  @a3/@x2
                @a1/@x3   @a2/@x3  @a3/@x3
\special {t4ht=
(2.3)

2.1.2 Divergence

If a vector field a  \special {t4ht= is defined and continuously differentiable then the divergence of a  \special {t4ht= is a scalar field

               @a    @a     @a
 \~/  • a = @iai =--1-+ ---2 + ---3
               @x1   @x2    @x3
\special {t4ht=
(2.4)

The divergence can operate on any tensor field of rank 1 and above to produce a tensor that is one rank lower. For example the divergence of a second rank tensor field T  \special {t4ht= is a vector field (expanding the vector as a column array for convenience)

                (                                   )
                   @T11/@x1 + @T12/@x1  + @T13/@x1
 \~/  •T = @iTij =    @T21/@x2 + @T22/@x2  + @T23/@x2
                   @T31/@x3 + @T32/@x3  + @T33/@x3
\special {t4ht=
(2.5)

2.1.3 Curl

If a vector field a  \special {t4ht= is defined and continuously differentiable then the curl of a  \special {t4ht=,  \~/   a  \special {t4ht= is a vector field

                   (                                   )
 \~/   a = e  @ a  =   @a3--  @a2, @a1-- @a3-, @a2 - @a1-
          ijk j k     @x2    @x3  @x3   @x1  @x1    @x2
\special {t4ht=
(2.6)

The curl is related to the gradient by

           *
 \~/   a = 2 ( skew  \~/ a)
\special {t4ht=
(2.7)

2.1.4 Laplacian

The Laplacian is an operation that can be defined mathematically by a combination of the divergence and gradient operators by  \~/ 2  =_   \~/  • \~/  \special {t4ht=. However, the Laplacian should be considered as a single operation that transforms a tensor field into another tensor field of the same rank, rather than a combination of two operations, one which raises the rank by 1 and one which reduces the rank by 1.

In fact, the Laplacian is best defined as a scalar operator, just as we defined nabla as a vector operator, by

             2      2     2
 \~/ 2  =_  @2  =_ -@--+  @---+  @---
           @x21   @x22    @x23
\special {t4ht=
(2.8)

For example, the Laplacian of a scalar field s  \special {t4ht= is the scalar field

  2     2    @2s-   @2s-   @2s-
 \~/  s = @ s = @x21 + @x22 +  @x23
\special {t4ht=
(2.9)

2.1.5 Temporal derivative

There is more than one definition of temporal, or time, derivative of a tensor. To describe the temporal derivatives we must first recall that the tensor relates to a property of a volume of material that may be moving. If we track an infinitesimally small volume of material, or particle, as it moves and observe the change in the tensorial property f  \special {t4ht= in time, we have the total, or material time derivative denoted by

df-=  lim  Df--
dt    Dt-->0 Dt
\special {t4ht=
(2.10)

However in continuum mechanics, particularly fluid mechanics, we often observe the change of a f  \special {t4ht= in time at a fixed point in space as different particles move across that point. This change at a point in space is termed the spatial time derivative which is denoted by @/@t  \special {t4ht= and is related to the material derivative by:

df    @f
---=  ---+  U • \~/ f
dt    @t
\special {t4ht=
(2.11)

where U  \special {t4ht= is the velocity field of property f  \special {t4ht=. The second term on the right is known as the convective rate of change of f  \special {t4ht=.