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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?

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It is customary to accept the most helpful answer, or edit your question to point out what is lacking so far. Your 0% accept rate is unacceptable here. To accept an answer you can click the transparent check sign by the voting arrows of the same answer. Remember that you can only accept one answer. –  Asaf Karagila Oct 13 '11 at 17:01
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@AsafKaragila +1 Moreover, some of his questions have received some very nice answers that required considerable effort on the part of the suppliers –  ItsNotObvious Oct 13 '11 at 19:35
    
@AsafKargila- Actually , I did not know about this. I am a bit new to math.stackexchange. Hence I am unaware of this custom to accept the answers. From the next time I will take care of it. –  Primeczar Oct 16 '11 at 19:20
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