Exploring Vector Analysis with Del Operator

In summary: This is the way I think about it, and I think it's the most correct. In summary, the del operator, represented as \nabla, is defined in 3-dimensional, cartesian coordinates as \nabla=\bold{i}\frac{\partial}{\partial x}+\bold{j}\frac{\partial}{\partial y}+\bold{k}\frac{\partial}{\partial z}. It serves as a mathematical tool for notation and has many implications in physical and mathematical equations. It is a function that can modify another function according to certain rules for convenience. Its use requires intuition and understanding of its purpose and implications.
  • #1
okay
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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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  • #2
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?
 
  • #3
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?
 
  • #4
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] [Broken]
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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  • #5
thank u defennnder
it made great sense.
 
  • #6
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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1. What is a vector?

A vector is a mathematical quantity that has both magnitude and direction. It is often represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow indicating the direction.

2. What is the Del operator?

The Del operator, also known as the nabla symbol, is a mathematical operator used in vector calculus to represent the gradient, divergence, and curl of a vector field. It is represented by the symbol ∇.

3. How is the Del operator used in vector analysis?

The Del operator is used to perform operations on vector fields, such as finding the gradient, divergence, and curl. It is also used in many physical laws and equations, such as Maxwell's equations and the Navier-Stokes equations.

4. What is the difference between the gradient, divergence, and curl?

The gradient of a vector field represents the direction and rate of change of the field, while the divergence represents the tendency of the field to spread out or converge. The curl represents the rotation of the field around a certain point.

5. Why is vector analysis important in science and engineering?

Vector analysis is essential in many branches of science and engineering, such as physics, engineering, and fluid dynamics. It allows for the analysis of complex systems and phenomena, and is used in the development of many mathematical models and theories.

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