Curl of a given vector field, $F$, at a given point is a vector, $C$, that gives the measure of amount of rotation the vector undergoes at that point. It has been defined using cross products and determinants. So the curl of a velocity vector is the angular velocity vector. so can I opine that curl of a vector field, $F$, gives a vector, $C$, such that position vector $\times C = F$?
If yes then can anyone provide me a derivation of the formula for curl (the vector with which when $R$ is crossed we obtain the vector whose curl we want to determine)?
Now similar is the case of divergence of a vector field, at a given point it provides us the amount of how much the vector spreads. Can anyone explain to me why it has been defined as the way it is, i.e. derivative of $x$ component wrt $x$ + derivative of $y$ component wrt $y$ + derivative of $z$ component wrt $z$, or provide me with its derivation if there is any?