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#1
Hi all!

My lecturer keeps saying that the divergence of a vector field and the curl of a vector field are both intrinsic- that they are independent on the co ordinate system. I understand how this can be so for divergence since it gives a scalar value but curl is a vector- what does it mean to be intrinsic in this case where different base vectors are used in different systems? does it mean they are equivalent if one were to use a relevant "conversion equation" linking the co ordinates of the new and old system ?

He only said "they have the same form" and I'm also not sure what he meant by form in this case

does this mean the del operator is intrinsic?

(Sorry for this mumbled mess I'm a bit confused, appreciate any help/clarification!)
(lol and I'm too embarrassed/shy/awkward.. to ask him directly )
Last edited by marinaelise; 1 year ago
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1 year ago
#2
(Original post by marinaelise)
Hi all!

My lecturer keeps saying that the divergence of a vector field and the curl of a vector field are both intrinsic- that they are independent on the co ordinate system. I understand how this can be so for divergence since it gives a scalar value but curl is a vector- what does it mean to be intrinsic in this case where different base vectors are used in different systems? does it mean they are equivalent if one were to use a relevant "conversion equation" linking the co ordinates of the new and old system ?

He only said "they have the same form" and I'm also not sure what he meant by form in this case

does this mean the del operator is intrinsic?

(Sorry for this mumbled mess I'm a bit confused, appreciate any help/clarification!)
(lol and I'm too embarrassed/shy/awkward.. to ask him directly )
Look at it like this. Changing the coordinate system doesn't change the vector (or vector field) itself, just its representation in the new coordinates. Divergence represents a sink or a source, in a vector field, which obviously doesn't change just because we choose to represent the coordinates differently. Similarly the curl represents the circulation, which isn't going to change just because you are using different coordinates. The numbers you get for the base vector components will be different, but the physical situation will be exactly the same.

You get the same physical answer in any coordinate system, regardless of whether del is written using rectangular, cylindrical, spherical, or whatever. It's just that the representation in each system is different, and some problems are easier to solve (due to symmetries) in certain coordinate systems.
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#3
(Original post by David Getling)
Look at it like this. Changing the coordinate system doesn't change the vector (or vector field) itself, just its representation in the new coordinates. Divergence represents a sink or a source, in a vector field, which obviously doesn't change just because we choose to represent the coordinates differently. Similarly the curl represents the circulation, which isn't going to change just because you are using different coordinates. The numbers you get for the base vector components will be different, but the physical situation will be exactly the same.

You get the same physical answer in any coordinate system, regardless of whether del is written using rectangular, cylindrical, spherical, or whatever. It's just that the representation in each system is different, and some problems are easier to solve (due to symmetries) in certain coordinate systems.
ok wow this couldn't be clearer! Sorry I feel a bit dim, I actually hadn't realised that, I guess I needed something to be said explicitly (like you have here), Thank you so much for this, had a bit of a revelation (and I'm so happy!)
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