2.1 Differential operators

Before defining the spatial derivatives we first introduce the nabla vector operator (from where Nabla Ltd. takes its name), represented in index notation as :

(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. ;
• otherwise the nabla operator behaves like any other vector in an algebraic operation.

2.1.1 Gradient

If a scalar field is defined and continuously differentiable then the gradient of , is a vector field

(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 is a second rank tensor field

(2.3)

2.1.2 Divergence

If a vector field is defined and continuously differentiable then the divergence of is a scalar field

(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 is a vector field (expanding the vector as a column array for convenience)

(2.5)

2.1.3 Curl

If a vector field is defined and continuously differentiable then the curl of , is a vector field

(2.6)

The curl is related to the gradient by

(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 . 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

(2.8)

For example, the Laplacian of a scalar field is the scalar field

(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 in time, we have the total, or material time derivative denoted by

(2.10)

However in continuum mechanics, particularly fluid mechanics, we often observe the change of a 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 and is related to the material derivative by:

(2.11)

where  is the velocity field of property . The second term on the right is known as the convective rate of change of .