Why the curl of a conservative force field is zero everywhere?
Because a field being conservative is equivalent to having zero curl. This should be derived in any basic text on vector analysis.
Can your conservative force be written as a gradient of a scalar field?
ummm this didn't help
Now take the curl of that gradient.
Ummm I'm looking for the physical meaning and significant of this
I am saying you should find this explained in detail in any basic textbook. This makes me wonder what effort you spent on trying to find the answer before posting.
This is mentioned without any illustration in my physics book, and there is nothing called conservative force in mathematics to explain it in a math book
Perhaps not conservative force, but certainly conservative vector field. A conservative force field is just a conservative vector field describing a force.
Well, if u can find that then plz send me a link
Loosely speaking, non-zero curl means that the vector field "goes in circles" somewhere, that you can follow the vector at one point to another and eventually get back where you started without ever going against the direction of the vector field at some point. For example, if the vector field were describing the current at the surface of a body of water, non-zero curl would mean that there was a whirlpool somewhere, so you could go around and around in circles without ever having to go against the current.
But a force field with that property cannot be conservative because you can return to your starting point with more or less energy than you started with, depending on whether you went with the current or against it.
That's the "loosely speaking" hand-waving picture that may help you visualize the physical significance of the math. However, the truth is in the math, so your next step is to go back to one of the many mathematical proofs that the curl of a gradient is zero, work through that proof now that you have a picture in your mind.
Ohhthanks for ur time and that sweet link
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