# Curl operator

The following are important identities involving derivatives and integrals in vector calculus. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity. Specifically, for the outer product of two vectors. When the Laplacian is equal to 0, the function is called a Harmonic Function. That is. Less general but similar is the Hestenes overdot notation in geometric algebra.

The dotted vector, in this case Bis differentiated, while the undotted A is held constant. We have the following generalizations of the product rule in single variable calculus.

We have the following special cases of the multi-variable chain rule. See these notes. The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connectionwhich differentiates a vector field to give a vector-valued 1-form.

The divergence of the curl of any vector field A is always zero:. This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex. The Laplacian of a scalar field is the divergence of its gradient:. Divergence of a vector field A is a scalar, and you cannot take the divergence of a scalar quantity. The divergence of a vector field A is a scalar, and you cannot take curl of a scalar quantity.

Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles dashed mean that DD and GG do not exist.

Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense analogous to interchanging the limits in a definite integral :.

Mean value theorem Rolle's theorem. Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem. Fractional Malliavin Stochastic Variations. Glossary of calculus. Glossary of calculus List of calculus topics. Main article: Gradient. Main article: Divergence. Main article: Curl mathematics. Main article: Laplace operator. The Feynman Lectures on Physics. Vol II, p.

Geometric algebra for physicists. Cambridge University Press. University of Auckland.Hot Threads.

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Search titles only. Search Advanced searchâ€¦. Log in. Contact us. Close Menu. JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding. Forums Physics Other Physics Topics. Is The Curl Operator Linear? Sheldon Cooper. Hi, I stumbled upon thinking that "Is curl operator a linear operator"? I was reading EM Theory and studied that the electromagnetic field satisfies the curl relations of E and B.

But if the operator was not linear then how can a non linear operator give rise to a linear solution. Thus it becomes apparent that curl is linear but how can we prove it mathematically? Thanks in advance. ShayanJ Insights Author. Gold Member. Does an operator have to be linear to generate a linear solution to an arbitrary equation, I did not know that?? A proof for the general case?

Does an operator have to be linear to generate a linear solution to an arbitrary equation?? I can't get the question, linear has multiple meanings in math. Insights Author. Linear can mean supper position, straight line, type of vector spacesomething raised to power one, a type of DEThe divergence and curl of a vector field are two vector operators whose basic properties can be understood geometrically by viewing a vector field as the flow of a fluid or gas.

Divergence is discussed on a companion page.

## Math Insight

Here we give an overview of basic properties of curl than can be intuited from fluid flow. The curl of a vector field captures the idea of how a fluid may rotate. The vector field indicates that the fluid is circulating around a central axis.

A rotating vector field. If the vector field is interpreted as velocity of fluid flow, the fluid appears to flow in circles. From the graph's original perspective i.

If you rotate the graph, you might see dots floating along the axis of rotation. These dots are representations of vectors of zero length, as the velocity is zero there. More information about applet. This macroscopic circulation of fluid around circles i. Although you fix the center of the sphere, you allow the sphere to rotate in any direction around its center point. The rotation of such a sphere is illustrated in the below applet.

The sphere should actually be really really small, because, remember, the curl is microscopic circulation. A sphere rotated by a rotating vector field. A sphere is embedded in a rotating fluid whose velocity is given by the vector field.

The middle of the sphere is anchored to a point originally, the originbut it is allowed to rotate with the fluid. The rotation of the sphere by the fluid is an indication of the curl of the vector field. You can move the point at which the sphere is anchored by dragging it with your mouse. As the curl is a vector, it is very different from the divergencewhich is a scalar.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. There are a number of posts on this site asking similar questions and some of them have been answered to my taste at least partially but none give a complete answer that I am satisfied with.

See links at the bottom of this question for a small selection of posts asking related or even the same questions. Some people will call this an operator, some will call it a vector, some will call it a vector operator, and some will adamantly claim that it is not properly anything at all and you shouldn't call it any of these things and you should just treat it as a "notational convenience". First I want to take issue with the final claim that it is purely a notational convenience.

I think it is more than just a notational convenience for the following reason. This is evidence that the symbol carries some sort of mathematical structure to it which should be able to be captured in an independent definition. To that end I'm interested in a coordinate free definition of this symbol. The definition I gave above relies on using the usual Cartesian coordinates above.

Can one exist? But at the same time it is an object which can be fed as an argument to a dot product together with a vector form a different space and return a scalar. The idea of two vectors sitting next to eachother made me think it might be some kind of rank 2 contravariant tensor but I think that may have been a stretch. I am happy to say we are working in 3 dimension. Ok, that is fine. We can work in a space that has a metric defined on it.

Even that is fine as long as we can work in coordinate systems such as cylindrical or spherical where the metric is still flat but no longer has a trivial component representation.

Coordinate transformation on del operator. In the same article one can find a coordinate-free formulathat can be taken as the definition of the exterior derivative. This observation essentially closes the question.

If you want tor restict yourself to this case, then I doubt that it is ever possible to find a pure coordinate-free way of expressing the quantities under consideration i. In other words, you are forced to deal with coordinates and the dimension-related tricks in order to handle these quantities. Coming back to the expressions in Wikipedia, notice that they use the Hodge star, but we have not received yet any convincing answer on how to give a coordinate-free definition for it.

This doubles my pessimism, but I can be wrong and overlook something important. Nevertheless, I find that this question and the other attempts to answer it are very insightful.This class represents a URL. It allows you to manipulate each element of the URL independently of the others whether parsing an existing URL string or building a string from scratch.

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Skip to main content. Contents Exit focus mode. Important This class and its members cannot be used in applications that execute in the Windows Runtime. Is this page helpful? Yes No. Any additional feedback? Skip Submit.Quoting the wikipedia definition of the curl vector operator :.

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space. At every point in the field, the curl of that point is represented by a vector.

The attributes of this vector length and direction characterize the rotation at that point. The devil in this definiton lies in the word infinitestimal. Let me show what I mean. Consider the vector field defined by. By looking at these images my first reaction was that this field is most certainly a rotational one. Imagine my surprise when I actually did the math and my intuition proved to be completely wrong:. How can it be that this plot corresponds to an irrotational field?

Well, it depends on which rotation you are referring to macroscopic vs. Imagine that this vector field describes the flow of water in a pool with a sink at the bottom that sucks the water out it. If we put a small ball on the surface of the water, then the ball may move in two distinct ways:.

These two opposite effects may cancel out as in our case and then the curl is zero. Please mind that the image above is drawn in a large scale. In reality the green circle is infinitestimal. Another way to look at curl is as the average circulation of a field in a region that shrinks around a pointi. Where is the green area in the image above, as it shrinks into a point.

Recall though that the curl is a vector, so the correct way to connect the above formula with the curl is:. Another way to attack the problem is by calculating the average circulation of the vector field around the point. For simplicity let us assume that we are working on a vector field in two dimensions plane :. Contents Introduction Another way to view curl Relation of curl with the angular velocity at some point First method Second method Introduction Quoting the wikipedia definition of the curl vector operator : In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.

Imagine my surprise when I actually did the math and my intuition proved to be completely wrong: How can it be that this plot corresponds to an irrotational field?

If we put a small ball on the surface of the water, then the ball may move in two distinct ways: The general rotation of the flow around the z-axis z-axis is perpendicular to your monitor in the counterclockwise direction, along the direction of the stream lines. Since the arrows of the field are longer the closer we are to the z-axis, the field tends to push the ball more strongly on the side closest to the z-axis, rather than the opposite side.

Another way to view curl Please mind that the image above is drawn in a large scale. Recall though that the curl is a vector, so the correct way to connect the above formula with the curl is: where is the normal vector to the point where we measure the curl.

First method Let us calculate the curl of : The component of is for brevity we write instead of : Recall though that and similarly. Therefore: Similarly it is and. Therefore the curl is twice the angular velocity: Second method Another way to attack the problem is by calculating the average circulation of the vector field around the point.

## Curl (mathematics)

For simplicity let us assume that we are working on a vector field in two dimensions plane : We use the parameterizationwith. Therefore: Therefore:.Before we can get into surface integrals we need to get some introductory material out of the way.

That is the purpose of the first two sections of this chapter. In this section we are going to introduce the concepts of the curl and the divergence of a vector. There is another potentially easier definition of the curl of a vector field.

### The meaning of curl operator

This is defined to be. Note as well that when we look at it in this light we simply get the gradient vector. Next, we should talk about a physical interpretation of the curl. The divergence can be defined in terms of the following dot product. We also have a physical interpretation of the divergence.

This can also be thought of as the tendency of a fluid to diverge from a point. The next topic that we want to briefly mention is the Laplace operator. The first form uses the curl of the vector field and is.

The second form uses the divergence. If the curve is parameterized by. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i.

Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. This is a direct result of what it means to be a conservative vector field and the previous fact. Show Solution So, all that we need to do is compute the curl and see if we get the zero vector or not.