What is the Del Operator in Vector Analysis and How Does It Work?

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Discussion Overview

The discussion centers around the del operator in vector analysis, exploring its definition, physical meaning, and mathematical implications. Participants express confusion about its origins and functionality, particularly in three-dimensional contexts, and seek clarification on its role in determining vector field properties such as gradient and curl.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the origin of the del operator and its effectiveness in revealing properties of vector fields like rotation and curl.
  • Another participant provides the mathematical definition of the del operator in three-dimensional Cartesian coordinates but seeks further clarification on the nature of the proof requested.
  • A different viewpoint suggests that the del operator has physical significance and does not simply appear without reason, implying that its mathematical results stem from its definition.
  • One participant describes the del operator as a mathematical convention that simplifies notation and enhances comprehension of equations.
  • Another participant explains that operators, including the del operator, modify functions according to specific rules, emphasizing its convenience and intuitive use.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of the del operator; some view it as a mathematical convention while others seek a deeper physical understanding. The discussion remains unresolved regarding the proof and foundational significance of the operator.

Contextual Notes

Participants express uncertainty about the del operator's origins and its implications in three-dimensional vector analysis, indicating a need for further exploration of its mathematical and physical foundations.

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hello every one,
i am working on vector analysis and i have come across this definition of del operator.i don't 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.
 
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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

[PLAIN said:
http://en.wikipedia.org/wiki/Del][/PLAIN]
Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember.

The above should clear your confusion up a little. It's simply a mathematical convention which has a lot of mathematical implications.
 
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thank u defennnder
it made great sense.
 
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.
 
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