Is The Curl Operator Linear?

  • #1
Hi,

I stumbled upon thinking that "Is curl operator a linear operator" ?
I was reading EM Theory and studied that the electromagnetic field satisfies the curl relations of E and B. But if the operator was not linear then how can a non linear operator give rise to a linear solution. Thus it becomes apparent that curl is linear but how can we prove it mathematically?

Thanks in advance.
 

Answers and Replies

  • #2
2,809
604
Use the definition ##\displaystyle (\vec \nabla \times \vec F)\cdot \hat n \equiv \lim_{A_n\to 0} \frac{1}{A_n} \oint_{\partial A_n} \vec F \cdot \vec{dr} ## (where ##A_n## is a surface with unit normal ##\hat n## and ##\partial A_n ## is its boundary curve.)!
 
  • #3
908
223
Does an operator have to be linear to generate a linear solution to an arbitrary equation, I did not know that??

A proof for the general case?
 
  • #4
2,809
604
Does an operator have to be linear to generate a linear solution to an arbitrary equation??

A proof for the general case?

The phrase "linear solution" seems meaningless to me. Linearity is a property of an operator/equation.
 
  • #5
908
223
I can't get the question, linear has multiple meanings in math.
 
  • #6
2,809
604
I can't get the question, linear has multiple meanings in math.
The OP has realized that Maxwell's equations are linear equations and so the curl operation is linear too. He just wants to prove that it actually is.
 
  • #7
fresh_42
Mentor
Insights Author
2021 Award
16,436
15,477
I can't get the question, linear has multiple meanings in math.
Which are? You can have linearity on the objects ##O## or in the arguments ##A##, but it's always the same condition:
 
  • #8
908
223
Linear can mean supper position, straight line, type of vector space , something raised to power one, a type of DE...
 
  • #9
fresh_42
Mentor
Insights Author
2021 Award
16,436
15,477
Linear can mean supper position, straight line, type of vector space , something raised to power one, a type of DE...
Which are all together just linear combinations of certain objects, or linear in the argument in case of scalar multiplication.
However, I have not the slightest idea what "type of vector space" means. Vector spaces are per definition a set of linear combinations.
There is no "multiple meaning" of linearity in math.
 
  • #10
908
223
Correction type of space eg vector space.

How is a linear equation eg y=2x+1the same as a linear first order DE or a linear operator?


Linearity has context.
 
  • #11
fresh_42
Mentor
Insights Author
2021 Award
16,436
15,477
Correction type of space eg vector space.

How is a linear equation eg y=2x+1
This is no linear function. It's called an affine transformation. To call it "linear" isn't exact. ##y(0) ≠ 0##

the same as a linear first order DE or a linear operator?
It is linearity on the considered objects, here differential operators which are a special case of linear operators.

Linearity has context.
Wrong. The objects or arguments it applies to have a context, e.g. the derivatives in the example above.
Linearity itself is a property independent of what you regard in a special case, it simply says that you have a rule for addition and for scalar multiplication - whether it's applied to a superposition or an operator. The meaning of "linear" does not change.
 
  • #12
908
223
I learned something, thanks.
 
  • #13
Hey Shyan, tried your procedure to prove the linearity problem but and am stuck can u please add a few more steps(if possible).
 
  • #14
2,809
604
Hey Shyan, tried your procedure to prove the linearity problem but and am stuck can u please add a few more steps(if possible).
Just put ## \vec F=\vec G+b \vec H ## in the definition!
 
  • #15
ok got it thanks!
 

Related Threads on Is The Curl Operator Linear?

Replies
1
Views
2K
  • Last Post
Replies
3
Views
2K
I
  • Last Post
Replies
1
Views
2K
Replies
1
Views
3K
  • Last Post
Replies
1
Views
667
Replies
1
Views
1K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
1
Views
794
Replies
3
Views
879
  • Last Post
Replies
1
Views
2K
Top