hello every one, i am working on vector analysis and i have come across this definition of del operator.i dont understand where does it come from but it works great to determine rotation curl gradient or other stuff of a vector field.can anyone tell me how we are getting this magical operator is there a proof about this? in 2-D i ve found great deal of explanation about vectors but in 3-D it is really complicated. i am looking forward to seeing your helps.
The "del" operator is just defined in 3 dimensional, cartesian coordinates as [tex]\nabla=\bold{i}\frac{\partial}{\partial x}+\bold{j}\frac{\partial}{\partial y}+\bold{k}\frac{\partial}{\partial z}[/tex] I don't really understand what proof you are looking for; could you expand on your question?
i just want to know is it something special that this three partial derivatives working great to reveal these gradiant curl ..etc .or is it some thing good looking thing that appear in these equations as we try to determine curl, rotation.. so on?
In my opinion, you are probably asking what does del mean physically. It can't possibly appear from thin air, and all the mathematical results follow from it. I suggest this would help you: http://en.wikipedia.org/wiki/Del The above should clear your confusion up a little. It's simply a mathematical convention which has a lot of mathematical implications.
Operator is a function that can modify another function according to some rule, anything you want. A derivative is an operator because it changes a function to a different one according to a certain rule. Del is "magical" because it was structured like that for our convenience. You use it with intuition whenever you like.