Is Curl a Measure of Circulation Tendency?

[LucasGB]

When studying the curl, one often finds the explanation that the curl is a measure of the tendency of a vector field to circulate around a given point. But this doesn't make much sense to me, since there's no clear way to measure "tendency"? What are the units of "tendency"? Wouldn't you agree that it is way more clear and precise to state that the curl is proportional to the tendency of the field to circulate around a given point?

I know this is a rather unimportant topic, but I'd like to hear your opinion. Thank you!

 

[CompuChip]

Even if you include the term "proportional", "tendency" is still a fuzzy word. It is just meant to clarify the mental picture of what curl physically means, for example in some flow. If you insist on rigour, you should revert to the mathematical definition. The formula is completely unambiguous, however it won't give you much insight into what it is.

 

[Phyisab****]

Tendency is obviously not a mathematically rigorous term. I think it's pretty clear what curl and divergence mean if you just look at a vector field with zero curl but non zero divergence, and a field with zero divergence and non zero curl. I.e. maxwell's equations for electrostatics.

 

[LucasGB]

So you guys are suggesting that there is no precise and rigorous verbal way to describe what divergence and curl are?

Funny thing is, this problem doesn't seem to exist with the gradient. It can be clearly stated to be a vector whose direction is that of the highest rate of change of the scalar function, and whose magnitude is that rate of change.

I wonder why there's a precise verbal definition for the gradient, and yet one cannot be achieved for the curl.

 

[arildno]

I wonder why there's a precise verbal definition for the gradient, and yet one cannot be achieved for the curl and the gradient.

Because "twice the local, instantaneous rotation rate of an infinitesemal fluid element" is not a concept in Average Joe's daily life.

 

[LucasGB]

Because "twice the local, instantaneous rotation rate of an infinitesemal fluid element" is not a concept in Average Joe's daily life.

Oh, that's what it is? But I think even that description is not rigorous, for it doesn't have to be a fluid. Please note I'm not trying to find a verbal description which is accessible or simple, merely a precise verbal description. What is the way to describe the divergence?

There's also the description that the curl is the circulation per unit area at a point. I think this is precise, right?

 

[Phyisab****]

The average joe is not taking vector calculus. I am only a senior undergrad, I want to know the answer -sarcasm please!

 

[LucasGB]

I don't understand what you're talking about...

 

[Phyisab****]

That was a response to arildno, I am just repeating your question.

 

[arildno]

There's also the description that the curl is the circulation per unit area at a point. I think this is precise, right?

Sure.

Along with that one, you can say that divergence at a point is the volum expansion rate per unit volume.