Thanks Matthew!
I find it disconcerting that pseudovectors–aka rotational vectors (such as curl)–are continually referred to as vectors, blurring the distinctions between the two types of construction and causing much confusion and heartache for the uninitiated. A true vector’s direction is determined straightforwardly; a rotational vector’s direction is a matter of convention. The two have different symmetries, different multiplicative properties, and so on. Especially on a site like yours that I depend on for clarity in visual thinking, I would love to see a clear distinction drawn between the two. I feel that, as time goes on, things like this lumping of really very distinct categories of things need to gradually fall by the wayside in order to foster deeper, more direct, and more intuitive understanding.
I am in Calculus III right now, and I found this post to be helpful in order to better understand curl intuitively. I really find the math in this class fascinating, as it seems to be the most practical of any math class I’ve taken so far. It’s really neat how we can model real life forces that on the surface seem so complex with relative ease! Thanks again for this post.
Chris, I have some sympathy for your view that the concept of vector used here is too general. But this is a site which tries (and succeeds!) to make numerous poorly understood concepts simple. To do that, the intricacies found when any mathematical idea is pursued at length have to be trimmed, so concepts such as co/contra variant vectors, bi-vectors, pseudovectors, and their extensions in topics such as tensors and n-dimensional geometries have to be ignored. For example, in a different field, atoms don’t really(?) look like minature solar systems, but the concept has been useful in teaching thousands of chemistry students.
Getting the foundations down, even if they’re not perfectly accurate, is the most important thing. Of course, cautions can be made that the model used is a model only, it’s useful only up to a point. Korzybski’s idea that the map is not the territory is apt - (remember AE Van Vogt’s “World of Null-A”?).
[…] Vector Calculus: Understanding Circulation and Curl | BetterExplained http://betterexplained.com/which is a cross-product of the gradient and the field (F). This has to do with how curl is actually computed, which will be … I'm terrible at math, I got a D in Calculus in high school, but I like the conceptual visualization of mathematical principles. You wrote: “However, in a field with curl (like a whirlpool), you can …… Getting the foundations down, even if they're not perfectly accurate, is the most important thing. Of course, cautions can be made that the model used is a … […]
(the paddle is vertical, sticking out of the water like a revolving door – not like a paddlewheel boat).
@khalid
Can you link me to an image so that i can understand it more clear.
Thanks
ok physical significance i understood.but i cannot relatate the output of nabla cross (vector field) with the physical meaning.how is it relate with circulation /flow for curl/divergence.
as example we use differential operator to differentiate.
and from geometrical explanation we comprehend that differentiation is rate of change.
but in case of curl/divergence we only talking about physical significance but no discussion about the relation between mathematical expression & the so called physical sigificance (circulation/flow etc)
Is the wall of a whirlpool vortex more dense than the surrounding medium?
Bro,
You made me fell in love with Calculus.
thanks a lot,
but I have a doubt, how is it work , the grad of a curl=0
and curl of a grad=0…?
@ Sanoosh:
You did a wrong question, but i can understand what you meant.
Actually, Curl of a Divergence is 0, and vice-versa.
Curl shows a rotation, along any axis, and basically it is a closed path, while Divergence is when something diverge from a center and not rotates.
So, if something rotating, that can’t diverge, therefore Curl of Divergence=0.
and in the same way, if something diverging, that can’t rotate, therefore Divergence of Curl=0.
Sorry!!
If something rotating, that can’t diverge, therefore Divergence of Curl=0.
and in the same way, if something diverging, that can’t rotate, therefore Curl of Divergence=0.
Wish I discovered this while I was an undergrad : (
Hi, My doubt is regarding the gradient on an equipotential surface. Without becoming zero why is the gradient vector, pointing perpendicular to the surface? I’m not understanding why travelling perpendicular will move in the direction of maximum. Please Explain.
Hi, Kalid!
Your explanation in math concepts really helped me a lot. However, about the curl and circulation part, there is still one point I can’t visualize with an intuition.
According to Stoke’s theorem(which has already been included by your “circulation density” concept I think):
$$\oint \textbf{F}\cdot d\textbf{\textit{r}}= \iint \nabla \times \textbf{F}\cdot \mathbf{n}\textit{d}\sigma$$
I always notice that the curl is dotted with a normal vector, which in my mind is to constitute the circulation density, which in turn is multiplied by a small area before being integrated to fulfill the whole circulation. Therefore, the direction of curl field of a vector field is just somewhere arbitrary instead of always in the direction of the unit vector of the surface, and actual circulation density is the projection of curl field on the unit vector! This is somewhat contrary to your explanation that the direction of curl of is normal to the surface.
So, can you help me with the visualization of the direction of the curl field?
Hi Yuyang, great question, I should to clarify the post.
The curl vector ($$\nabla \times \textbf{F}$$) is a property of the vector field, and points along the axis of rotation. This direction is independent of any surface that is later placed in the field.
The surface orientation at a tiny area is the normal vector n, and the dot product measures how much of the curl’s rotation would be captured by that small surface area element. Stokes Theorem says we add up all these “captured rotations” to get the full circulation around the boundary.
The curl vector can be in an arbitrary direction, because it’s based on the vector field, not the surface we place. (As a thought experiment, you can have many different surfaces in the same vector field, and each captures a different amount of rotation.) Hope that helps!
Thanks a bunch… Helped me ALOT…