Originally Posted by **DesertFox** Hello, guys!!
Does an __infinitesimal change in __**D** (electric displacement field) change __the rate of change of __**l** with **x** ?
If the answer is YES, can you, please, write the correct mathematical expression of this? |

First I'd like to make you aware of the other name that

**D** is referred to in the physics literature, i.e.

**Electric Flux Density**. In some texts

**D** is not given a special name but merely referred to as an auxiliary field.

Now, regarding your question, it doesn't make any sense to me. In the first place you didn't define the terms that you used, i.e. what are the quantities

**l** with

**x**. Since you didn't define them I'll have to guess what they are.

I'll assume that

**l** is current and

**x** is the position vector. Is that correct? If so then I don't see a direction relationship between these quantities as a general rule. For example: let's say we have a material which is, in all generality represented by a non-linear combination of the components of the electric field. The problem with your question is you wrote

"infinitesimal change in

**D**" without telling us whether its a change due to an infinitesimal change in time or an infinitesimal change in position.

Nor did you mention the same thing with respect to the current.

If you meant does the change in D in position change dI/dx then the answer is not well defined since it will depend on the exact situation. This is because there can be a non-zero D with no current present at all such as an EM wave in space. Or you can have a current in a material which has a non-linear relationship between D and E. But I see no relationship between that and dI/dx whatsoever.

So your question makes no sense to me. Why do you ask?