In

continuum mechanics a

material line element is a small

displacement vector d

**l** between two

particles located at

**x**,

**x**+d

**l**. d

**l** is "built into" the medium. As the

medium is

deformed the displacement between the particles changes, so d

**l** depends on

time.

The main reason why we are interested in material line elements is that we for many purposes (especially integration) like to think of curves as being approximable by a chain of small displacement vectors. If we know what happens to the material line elements we therefore also know something about the behaviour of curves that are fixed in the medium.

For this reason we wish to find the precise form of the time-dependence of d**l**. Consider the change in d**l** during a short time dt. In this time the position of the particles will have changed to **x** + **u**(**x**)dt, **x**+d**l** + **u**(**x**+d**l**)dt, respectively, where **u** is the velocity field of the medium. Therefore to linear order

d(d**l**) = ((**x**+d**l** + **u**(**x**+d**l**)dt) - (**x** + **u**(**x**)dt) - d**l**) = **u**(**x**+d**l**)dt - **u**(**x**)dt = (d**l**.**∇**)**u**dt

Using the substantial derivative operator D/Dt to indicate that we move with the medium while we measure the change in d**l** we can write this as

Dd**l**/Dt = (d**l**.**∇**)**u**

The interpretation of this is that if **u** increases in the direction of d**l** then d**l** will be stretched out by the motion of the medium.