What is the meaning of the sufficient condition?
An example, When we say the suffiecient condition for a vector field to be conservative is the curl of it equals zero.
So, if we have a conservative vector field, then its curl will be zero.
But does that mean that if the curl of the vector field is zero, then it is conservative?
Can we generalize this to any sufficient condition relation?
A condition being "sufficient" just means that it is enough to imply some other property. Symbolically, we say "$P$ is a sufficient condition for $Q$" or "a sufficient condition for $Q$ is $P$" both mean the same thing as "$P$ implies $Q$", or "$P\to Q$." The order here is quite important, as generally $P\to Q$ is not equivalent to $Q\to P$. To apply to your example, we do have that for a vector field, if it is conservative, then it has zero curl. In the language of sufficiency, we would say that "A sufficient condition for a vector field's curl to be zero is that it is conservative." The order is important there, as the converse, "If a field has zero curl, then it is conservative," is false (see for example this answer: Does zero curl imply a conservative field?). A closely related word sometimes also used in proofs is "necessary," usually written "$Q$ is a necessary condition for $P$" meaning the same thing as $P\to Q$.
