Question about the formal definition of conservative force

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Main Question or Discussion Point

So, I was reading the mathematical description of a conservative force o wikipedia : https://en.m.wikipedia.org/wiki/Conservative_force and at the line "Many forces (particularly those that depend on velocity) are not force fields. In these cases, the above three conditions are not mathematically equivalent".
Why these statements are not mathematically equivalent?
 

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vanhees71
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