Vector Calculus: Understanding Divergence & Curl

[Isaac0427]

Hi! I have recently been independently studying vector calculus. I understand that divergence measures change in magnitude and curl is the change in direction, however, I don't understand what certain divergences and curls represent. For example, how would you describe a field with a divergence of 2? What about a curl of 2? What about when divergences and curls have veriables in them (i.e. 3x+7y)? Thanks in advance!

 

[Nugatory]

Hi! I have recently been independently studying vector calculus. I understand that divergence measures change in magnitude and curl is the change in direction, however, I don't understand what certain divergences and curls represent. For example, how would you describe a field with a divergence of 2? What about a curl of 2? What about when divergences and curls have variables in them (i.e. 3x+7y)? Thanks in advance!

Curl is a vector and 2 isn't a vector, so saying that a field "has a curl of 2" is meaningless.

If you see variables such as $x$ and $y$ in the expression for divergence that means that the divergence is different at different points in space; $x$ and $y$ are the coordinates of the point where you're calculating the divergence. The same thing works for the expressions for the components of the curl vector.

If your vector field is the current in a body water (at every point $(x,y)$ the water is moving in some direction with some speed) the divergence will be non-zero at the inlets and outlets where water is entering or leaving the system, and the curl will be greatest in the neighborhood of the whirlpools (if any).

If any part of what I just said does not make sense, you need to go back and solidify your intuitive understanding of what a vector field is.

 

[Isaac0427]

Curl is a vector and 2 isn't a vector, so saying that a field "has a curl of 2" is meaningless

I was using http://www2.sjs.org/raulston/mvc.10/topic.6.lab.1.htm and I just want to know visually, what it means for those examples to have the curl and divergence that they are said to have.

 

[Nugatory]

I was using http://www2.sjs.org/raulston/mvc.10/topic.6.lab.1.htm and I just want to know visually, what it means for those examples to have the curl and divergence that they are said to have.

Look halfway down that page and you'll see that it says "Now, technically, since curl represents a direction, it has to be represented by a vector, and it’s more correct to say that

. If you’re conversant with torque, you know that it acts in a direction perpendicular to the plane of the forces causing the spin, so it wouldn’t surprise you that curl G is in the z-direction for Field 10."