Help with the conventions used for curl operator

In summary, the conversation is about a person trying to understand the derivation of curl on the given website. They are confused about the notation and why the line integral is taken in the Δz direction when the change is in the y direction. The expert explains that the component Ay is integrated left-to-right along the bottom of the square and also along the top of the square from right-to-left, and that the person has now understood the explanation.
  • #1
ppoonamk
28
0
http://web.mit.edu/6.013_book/www/chapter2/2.4.html

I was going through the curl derivation on the above link. How does equation 3 turn out? Δy is the incremental length. But how do you decide whether it is +Δy/2 or -Δy/2. And why is the line integral taken Δz when the change is in the y direction. I am utterly confused :(
 
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  • #2
ppoonamk said:
http://web.mit.edu/6.013_book/www/chapter2/2.4.html

I was going through the curl derivation on the above link. How does equation 3 turn out? Δy is the incremental length. But how do you decide whether it is +Δy/2 or -Δy/2. And why is the line integral taken Δz when the change is in the y direction. I am utterly confused :(

The component Ay is integrated left-to-right along the bottom of the square (where z = -Δz/2, and y goes from -Δy/2 to +Δy/2), and also along the top of the square from right-to-left.

RGV
 
  • #3
Sorry I still did not understand you properly :( Please forgive my ignorance about this. What is
+Δy/2 and -Δy/2. Why is the entire Ay compnonent have a 1 sign before it. AT the bottom of the sqyare on the left z is -Δz/2 but the first component of Ay has +Δz/2 term
 
  • #4
I understood. Thank you :)
 

1. What is the curl operator?

The curl operator is a mathematical vector operator that calculates the rotation or "curl" of a vector field at a given point in space. It is represented by the symbol ∇ x and is often used in physics and engineering to describe the behavior of fluid flow or electromagnetic fields.

2. How is the curl operator used in vector calculus?

In vector calculus, the curl operator is used to calculate the circulation of a vector field around a closed path. This is also known as the line integral of the vector field. The value of the curl at a point represents the tendency of the vector field to rotate around that point.

3. What are the conventions used for the curl operator in mathematics?

In mathematics, the curl operator is typically represented by the symbol ∇ x, where ∇ is the nabla symbol and x is a cross product. This notation is used to denote the curl of a vector field, and is often used in equations and formulas involving vector calculus.

4. How is the curl operator used in fluid dynamics?

In fluid dynamics, the curl operator is used to describe the behavior of fluid flow, specifically the vorticity or rotation of the fluid at a given point. The curl of the velocity vector field can help determine the direction and strength of the flow at a particular point, which is important for understanding turbulence and other fluid phenomena.

5. Are there any other applications of the curl operator?

Yes, the curl operator is also used in electromagnetism to describe the behavior of electromagnetic fields. It can help determine the strength and direction of magnetic fields, and is an important tool in understanding electromagnetic induction and Maxwell's equations.

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