curl ( mathematics )
calculus - vector field, curl and divergence. simple computing and proving: - mathematics stack exchange
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smith, d. e. history of mathematics, vol. 2: special topics of elementary mathematics. new york: dover, p. 329, 1958 .
, then the operator will clear the image to black, due to the mathematics involved .
why is does this vector field have zero - curl everywhere? plus, broad questions about curl .
free ebook urltoken i present and solve a simple example where the curl of a given vector field is sought. the curl is one of the basic operations of
vector calculus
. such and example is seen in 2nd year university mathematics. such an example is seen in 2nd year university mathematics courses .
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strang points out in his cse book that div curl = 0 is the transpose of curl grad = 0 .
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nykamp dq, “the idea of the curl of a vector field. ” from math insight. urltoken / curl _ idea
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so, though the curl near the red point is negative, the curl in the neighborhood of the blue point is positive .
can you show how to derive the identity curl x curl (f) = del (del. f) – delsquared f ?
related question: what are the eigenvectors of the curl operator? which might be used to construct $ \ curl $ from its eigenvectors .
for some idea of what curl does to a field; in a whirlpool or a tornado all of the curl is in the center funnel, and the field (wind) wraps around where the curl is .
now, let' s take more examples to make sure we understand the curl. what can we say about the curl of the vector field
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from green' s theorem we know that the curl of gradient field must be zero. we will use that to get formula for curl in space
this lecture gently introduces the idea of a the curl of a vector field. the curl is one of the basic operations of vector calculus. dr chris tisdell discusses the definition of the curl and how to co. .
curl (mathematics) in vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3 - dimensional vector field. at every point in the field, the curl of that point is represented by a vector. - video is targeted to blind users attribution: article text available under cc - by - sa image source in video urltoken
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a programming environment for developing rich internet applications (rias) from curl, inc. , cambridge, ma (www. curl. com). conceived at mit, curl combines html markup with an object - oriented programming language. the user' s machine requires the curl runtime engine and browser plug - in for execution. the first curl implementation was released in 2002. see
conservative fields have zero curl: there are no free twists to push you along. alternatively, if a field has curl, it is not conservative .
this theorem about the divergence of the curl is responsible for the conservation of charge. begin with maxwell' s equation for the curl of the magnetic field
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this is great. it makes it easy for me as a lecturer of mathematics to explain to my student teachers who will in turn explain the concepts well to their pupils. i would also appreciate if you could send me some similar ideas on complex variables and any other areas of mathematics .
if the paddle does turn, it means this field has curl at that point. if it doesn' t turn, then there' s no curl .
i understood divergence after reading feynman lectures. it' s just the curl .
( that is, we want to know if the curl is zero) .
the span consists of four little bridges that curl up to form two hearts .
curl of a vector field (ex. no. 1): vector calculus
curl of a vector field (ex. no. 2): vector calculus
then just use the fact that the curl of the gradient is identically zero .
we first calculate the necessary partial derivatives for the curl formula. we have that
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we have a couple of nice facts that use the curl of a vector field .
we also have the following fact about the relationship between the curl and the divergence .
i am not sure if there is an analogy between torque and curl. @ adityadev
curl of vector quality indicates that how much the vector quality curls or twist around .
the curl is a vector that measures the rotation or circulation of a vector field .
curl: stick a weightless toy vane to be centered at x. how fast does the vane rotate? the amount of rotation is the amount of curl at x .
i think curl is a little more abstract, this helped me understand it better .
these muscles work together to curl the toes down from the first and second joints .
if your hair curls or if you curl it, it is full of curls .
for (c), look at various formulas you know about div and curl .
$ \ nabla \ wedge e $ is what differential forms people call the curl .
is important in remembering the formula for the curl of a vector field. recall that
let' s look at some examples of determining the curl of a vector field .
we will first calculate the necessary partial derivatives for the curl formula. we have that
curl tells you how much vector quality is spinning (curling) around a point. think of rotating water in a bucket has curl. you can measure curl by putting a piece of dust in the liquid and seeing if it spins around its own axis .
curl is
circulation density
or circulation / area. shrink the hula hoop to a cheerio. if it' s still twisting, there is curl at that point .
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so, the curl isn’t the zero vector and so this vector field is not conservative .
thanks a lot. . i am a million times clearer about what a curl is now… but i still have a doubt - why should the curl of conservative field be zero. cant a conservative field have unsymmetrical distribution of vectors which can lead them to have curl ?
an object round which hair is rolled to make it curl, fastened in the hair .
flux of vector field across surface vs. flux of the curl of vector field across surface
please do a quora search on divergence curl. you will find many, many answers .
curl of a vector field (ex. no. 1): vector calculus - youtube
we must now make things more complicated. is the curl of figure 2 positive or negative, and in what direction? because we are observing the curl that rotates the water wheel in the x - y plane, the direction of the curl is taken to be the z - axis (perpendicular to plane of the water wheel). in addition, the curl follows the right - hand rule: if your thumb points in the + z - direction, then your right hand will curl around the axis in the direction of positive curl. for figure 2, the curl would be positive if the water wheel spins in a counter clockwise manner. the curl would be negative if the water wheel spins in the clockwise direction .
these equations are first order, which usually means the mathematics will be easy (good !), but they' re also coupled, which means they really aren' t (rats !). let' s separate them using this little trick. take the curl of both sides of faraday' s and ampère' s laws. the left side of each equation is the curl of the curl, for which there is a special identity. the right side of each equation, on the other hand, is the curl of a time derivative. we' ll switch it around into a time derivative of the curl .
@ tobiaskienzler i' ve added one last section on the translation of your identities for the curl to geometric calculus. i think this bears specifically on your search for a generalized curl .
recall that by convention (a bunch of people agreeing), counterclockwise circulation will give a curl pointing out of the page. using these facts, we can create the formula for curl :
you' ll see fancier equations for curl where the surface shrinks to zero (such as in wikipedia), but recognize the basic intuition - - curl is the circulation per unit area .
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 .
in colder weather, your cat will curl up into a tight, heat - conserving ball .
( of a vector or vector function) having curl equal to zero; irrotational; lamellar .
you only need this and the product rule to expand the double curl. it comes out to
, and the rate at which the object turns depends on the length of the curl vector .
the core of the mathjax project is the development of its state - of - the - art, open source, javascript platform for display of mathematics. our key design goals are
the mathematics behind all this is that we can estimate probabilities of players changing approach. since these are multi - dimensional problems, we can draw vector fields representing the probable outcomes .
( 1852–1914). poynting' s derivation involves vector mathematics that isn' t appropriate for the level of this book. (translation: i don' t understand it. )
i' m not confused about what curl is on a general level, i' m confused about why the formula for curl coincides with the physical interpretation, on an intuitive and rigorous basis .
if w is the way the wind is blowing, then the tornado is the curl of w. curl is the
twistiness
of the field, which points around the curl. although the wind around the tornado is moving, all of the twisting is happening only at the tornado itself .
we can draw the vector corresponding to $ \ curl \ dlvf $ as follows. we make the length of the vector $ \ curl \ dlvf $ proportional to the speed of the sphere' s rotation. the direction of $ \ curl \ dlvf $ points along the axis of rotation, but we need to specify in which direction along this axis the vector should point. we will (arbitrarily ?) set the direction of the curl vector by using the following “right hand rule. ” to see where $ \ curl \ dlvf $ should point, curl the fingers of your right hand in the direction the sphere is rotating; your thumb will point in the direction of $ \ curl \ dlvf $. for our example, $ \ curl \ dlvf $ is shown by the
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weisstein, eric w .
curl .
from mathworld - - a wolfram web resource. urltoken
1. conservative fields have zero curl. 2…. consider a river: it’s field is conservative .
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 .
for your question, think of a mixed region with non zero divergence and non - zero curl. the direction of the curl would be along the axis of rotation and hence perpendicular to the surface .
if the paddlewheel will turn, then the field has a non - zero curl at that point .
you don’t even have to have direction changes anywhere for a field to have nonzero curl, though .
geometrically, the curl of a vector field can be understood as follows. suppose that the vector field
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hi kalid i just wanted to know that since the curl is given by circular force / area……. will the surface area of an object dipped in the flowing water vary the curl experienced by it. that is will a smaller object with a smaller area experience greater curl than a larger object? this question came to me because i somehow feel that curl is analogus to pressure. am i going wrong somewhere ?
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therefore a river must have zero curl. if that is true, based on the two aforementioned premises .
( x, y, z), and we would like to determine the curl. the vector field
in the figure below, a has a clockwise curl around it, while b has an anticlockwise curl, indicating that the velocity vector is lower between a and b than on either side of a and b .
this video is regarding the' curl' and there are 2 other videos explaining about divergence and gradient .
is the curl positive, negative or zero in figure 4? and in what direction is it? first, since the water wheel is in the y - z plane, the direction of the curl (if it is not zero) will be along the x - axis. now, we want to know whether the curl is positive (counter - clockwise rotation) or if the curl is negative (clockwise rotation) .
so the curl is a measure of the rotation of a field, and to fully define the 3 - dimensional rotation we get a 3 - dimensional result (the curl in equation [ 3 ]). let' s look at a mathematical example of a vector field and calculate the curl. suppose we have a vector field
curl one' s lip, to raise a corner of one' s lip, as in showing disdain .
this gives about all the information you need to know about the curl. the important points to remember are that the curl operates on a vector function, and returns a vector function. the resulting curl is a measure of the rotation of the field in the 3 principal axis (x -, y -, z -) .
if you can imagine a rotating fluid, use the right hand to curl your fingers in the direction of the rotation of the fluid. your thumb will give you the direction of the curl vector for that flow field .
in geometric algebra and calculus, the cross product and curl are generalized by the wedge product and exterior derivative .
scalar and vector fields; gradient, divergence, curl, laplacian; cylindrical, spherical, orthogonal curvilinear coordinates .
what a great explanation. i can tell you my class has learned all about curl and divergence but very few of my classmates right now can conceptually understand what a curl really is. my book explained the whole curl concept in a paragraph and my teacher also gave a brief explanation because he never fully understood it in the first place .
so all that we need to do is compute the curl and see if we get the zero vector or not .
the presentation of the idea of curl via pictures does come with an important warning. one must remember that pictures can be deceiving. in particular, one might incorrectly think that the curl is due to the macroscopic circulation of the vector field that one readily notices in the above pictures. be forewarned that this assumption is far from the truth! if you think you might get confused between the microscopic circulation of the curl and the macroscopic circulation that one can observe from a picture, we urge you to check out some subtleties about the curl that may help you really understand curl .
$ t _ 1 * d t _ { 1 } ^ { - 1 } v $ is the curl .
first and foremost, while curl and divergence are measures of vector field, gradient is a measure of scalar field .
what is the difference between a curl, divergence and a gradient of a function? along with their physical significance .
it' s probably easier to understand it if you think of curl and divergence in terms of circulation and flux .
all the vector fields you’ve seen so far have had zero curl. it’s time to look at some that don’t .
this imbalance of forces leads to a torque and thus to the wheel’s spinning counterclockwise and hence to a positive curl .
i am not yet sure how to express $ \ delta $ and therefore can' t check whether $ \ curl ^ 2 = \ grad \ div - \ delta $. since $ d _ { i + 1 } \ circ d _ i = 0 $, one obtains $ \ curl ^ 2 \ grad = 0 $ and $ \ div \ curl ^ 2 = 0 $, which is necessary (but not sufficient) for the criteria $ \ curl \ grad = 0 $ and $ \ div \ curl = 0 $ from the question. these require
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this is exactly how i learned about div and curl in my physics course on electromagnetic theory, where they talk about the integral form of maxwell' s equations (which we call stokes' theorem) and take a limit to derive the differential form of maxwell' s equations (containing the div and curl). although i generally do not like the way physicists present mathematics, this is one case, where i think the physicists do it better .
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which is precisely equation (4). a somewhat more elegant formulation of the curl is given by the matrix operator equation
in this section we are going to introduce a couple of new concepts, the curl and the divergence of a vector .
where (s) is the surface we are considering; the direction of the curl is the normal to the surface .
( curl) a command line utility for executing functions with url - oriented protocols such as ftp and http. pronounced
why do you obtain curl on taking the determinant? i understood the $ \ vec { \ nabla } $ part .
this page on the curl (curl or rotation operator) is copyrighted, particularly the application to maxwell' s equations. no portion can be reproduced except by permission of the author. copyright maxwells - equations. com, 2012 - 2016 .
to me, the explanation for the appearance of div, grad and curl in physical equations is in their invariance properties .
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notice that it has both
counterclockwise curling
and
outward streaming
and hence both positive curl and positive divergence .
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curl tells you how much stuff is spinning (curling) around a point. rotating water in a bucket has curl. you can measure curl by putting a piece of dust in the liquid and seeing if it spins around its own axis. (although, to confuse you, a whirlpool doesn' t have curl. put a speck of dust in a whirlpool, and as it spirals down the drain, if you watch it closely it will not spin about its own axis. )
curl is simply the circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point). imagine shrinking your whirlpool down smaller and smaller while keeping the force the same: you' ll have a lot of power in a small area, so will have a large curl. if you widen the whirlpool while keeping the force the same as before, then you' ll have a smaller curl. and of course, zero circulation means zero curl .
in the definition that ‘curl is simply the circulation per unit area’, it will be more accurate to say that curl is simply the circulation per unit volume or the infinitesimal rotational field / force per unit volume, as i read in most places .
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the helix .
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i have trouble in understanding the above passage. i feel the determinant equivalent of the curl of v to be wrong, because we don' t get the curl of v when we solve that determinant. is it right to write in determinant form ?
we could repeat the procedure for maxwell' s equation for the curl of the electric field. there, we' d get
you might think from these examples that all vector fields with a non - zero curl show this “curling around a point” behavior .
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 ?
1) curl is a vector describing
instantaneous rotation
. the line integral over a gradient vector field is zero on any closed curve, so whatever instantaneous rotation is, it should have the property $ curl (\ nabla f) = 0 $ .
as far as explaining d ^ 2 = 0: curl (grad f) = 0 because the line integral of a gradient about small circles is zero, so gradients can' t curl. (in other words, if a vector field has nonzero curl at some p, you wouldn' t be able to define a consistent potential about some small closed contour around p .) and div (curl f) = 0 because the surface integral of a curl about small spheres is zero, so curls can' t diverge (that is, flow). these interpretations get used all the time in applications of stokes' theorem to physics .
( project solely managed by dr. paul ledger): electromagnetic scattering using a plane wave h (curl) conforming finite element method
as you can imagine, the curl has x - and y - components as well. hence, the curl operates on a vector field and the result is a 3 - dimensional vector. that is, if we know a vector field then we can evaluate the curl at any point - and the result will be a vector (representing the x -, y - and z - directions) .
if i have been successful, you should understand intuitively what circulation and curl mean, and how we got the formulae above. they spring up naturally from our definition of circulation as
pushing force along a path
and curl as
pushing force / area
.
there is another (potentially) easier definition of the curl of a vector field. to use it we will first need to define the
so what’s the units of measure when you get an answer to the “curl”. for instance in your whirlpool example, or in magnetism ?
i think that i understand the concept of the curl, i recommend this if anyone is interested. the problem is in the formula .
the specifics of the physical interpretation of curl (is it infinitesimal torque, average infinitesimal torque, average infinitesimal angular acceleration, etc. )
2. now move that charged particle, then it would generate magnetic field. curl is the magnetic field generated by that moving particle .
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the curl is a three - dimensional vector, and each of its three components turns out to be a combination of derivatives of the vector field $ \ dlvf $. you can read about one can use the same spinning spheres to obtain insight into the components of the vector $ \ curl \ dlvf $. the formula for the curl components may seem ugly at first, and some clever notation can help you remember the formula. once you have the formula, calculating the curl of a vector field is a simple matter, as shown by this example .
we can calculate $ \ mathbf { curl } \, \ mathbf { e } $ by $ d (e ^ 1dx _ 1 + e ^ 2dx _ 2 + e ^ 3dx _ 3) $. but how to calculate $ \ mathbf { curl } \, \ mathbf { curl } \, \ mathbf { e } $ using differential forms? i know the first $ \ mathbf { curl } \, \ mathbf { e } $ needs to be contracted with a tensor, but which tensor is it here: the metric ?
the' mathematics' compose method was added to im version 6. 5. 4 - 3. and is currently only available using the
- compose
operator. it is not available from the
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command due to its need for arguments .
why, here, is the area $ | a | $ the limiting factor? that doesn' t make much sense to me. is curl then the average infinitesimal torque, averaged over an infinitesimal area (since $ | a | \ to 0 $)? wikipedia also says that the curl is the
circulation density
, rather than simply the torque. so is my initial interpretation of the curl wrong ?
curl is related to circular field, the field along a current carrying conductor is circular it can either clock wise or anti clock wise depending on the direction of current flow. if the current is steady (constant) and wire is straight the curl is zero because field is uniform .
note to answerers: i declared the operators \ curl, \ grad and \ div for convenience, they should work everywhere below the question .
phd (dr. rer. nat .) awarded from department of mathematics, university of augsburg. thesis on '` multilevel preconditioning for mixed finite elements in three dimensions'', under the supervision of prof. dr. r. h. w. hoppe
the relationship between curl and the wedge derivative is through hodge duality. this is denoted by multiplication with a pseudoscalar, called $ i $. $ i $ converts vectors to 2 - vectors (
pseudovectors
) and vice versa. the canonical relationship between the curl and this derivative is
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@ swetha: for a field to be conservative, it means there is “no free ride” and therefore every loop you take has no “overall push”. if this is the case, it means curl (circulation / area) is zero at every point. a non - conservative field could have zero curl at some point, and non - zero curl at the others (it’s the non - zero areas which make it non - conservative) .
take a look at this railscast. in the show notes, you can see examples of using curl to grab a page with cookies and etags .
11. (mathematics) maths also called: rot or rotation a vector quantity associated with a vector field that is the vector product of the operator ∇ and a vector function a, where ∇ = i ∂ / ∂ x + j ∂ / ∂b y + k ∂ / ∂ z, i, j, and k being unit vectors. usually written curl a, rot a. compare divergence 4, gradient 4
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i am looking for the translation of
rotationnel
. it is a differential operator. in french, we say, for example ,
rotationnel de f
. i found two words :
curl
and
rotor
. is it correct? do we say
curl f
?
the vector field $ \ dlvf $ determines both in what direction the sphere rotates, and the speed at which it rotates. we define the curl of $ \ dlvf $, denoted $ \ curl \ dlvf $, by a vector that points along the axis of the rotation and whose length corresponds to the speed of the rotation. (as the curl is a vector, it is very different from the divergence, which is a scalar. )
ziegler, günter m. lectures on polytopes. vol. 152. springer, 2012 (ch. 0, 2, 7, 8) beck, matthias, and robins, sinai. computing the continuous discretely, second edition, springer, 2015. barvinok, alexander, lattice points, polyhedra, and complexity, ias / park city mathematics series, 2004. stanley, richard p. , two poset polytopes, discrete comput. geom. 1: 9 - 23 (1989). postnikov, alexander, permutohedra, associahedra, and beyond, international mathematics research
the final topic in this section is to give two vector forms of green’s theorem. the first form uses the curl of the vector field and is ,
what i am really hoping for is some way to convey to them that curl is in fact as natural as gradient and divergence, despite initial appearances .
things here are quite complicated, but to simplify just a bit, look in the immediate neighborhood of the red point. what you see there are positive divergence and negative curl. every one of the nearby vectors points outward, so there is no question about the divergence’s being positive. as for the curl, notice that all of the closest vectors are rotated clockwise from pointing straight out, so there is no question about the curl’s being negative either. .
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the 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. here we give an overview of basic properties of curl than can be intuited from fluid flow .
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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 .
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gerald, your remark seems rather harsh. i take my teaching responsibilities seriously, and i think that a phd in mathematics is sufficient qualification for anyone to pick up concepts from freshman / sophomore physics as needed. i would certainly not be pleased to receive such an inquiry from my department head .
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basically, i think it' s reasonable to tell people that curl is measuring curl, and that it should really be fed two vectors. in the plane, you don' t have to tell it the two vectors, and in 3 - space you can cheat differently by feeding it the third, orthogonal vector .
my question is, how can you place a paddle wheel in a river and have it turn, since is has zero curl? thanks for your answer. rocky
the curl of a vector function is a vector which is formally the cross product of the del operator and the vector. also known as rotation (rot) .
in figure 2, we can see that the water wheel would be rotating in the clockwise direction. hence, this vector field would have a curl at the point
div, grad, curl, and all that: an informal text on vector calculus (fourth edition): h. m. schey: 9780393925166: urltoken books
first of all, curl is just a utility for performing http requests. it doesn' t automatically record response headers such as the cookie or the etag header .
for (b), use stokes' s theorem. what is $ \ text { curl } \, \ mathbf f \ cdot \ mathbf n $ ?
very good start. please make it more rigorous by introducing the concept of gadient, divergence and curl. thus, please explain connotations of maxwell equations vectorially. sirsindhi
( with gisela widmer, sam, and prof. dr. jun zou, the chinese university of hong kong): auxiliary space preconditioning in h (curl )
falling more under the heading of
surprising connection
is the relationship between the determinant and the volume and thus, by cramer' s rule, the association between the solution of a system of linear equations and volume. indeed, the determinant manages to creep its way into a surprising amount of mathematics .
then, curl describes circulation though an infinitesimally small path, and divergence describes flux through an infinitesimally small surface. in other words instead of depending on a closed path or surface, it' s dependant on a point in the vector field (you would talk about the curl or divergence at a point in the vector field) .
by becoming a mathjax friend, organizations show the community that they support the goal of easy - to - use, high - quality mathematics display on the web for everyone, and are contributing in a very concrete way to help mathjax realize that goal. mathjax friends will be listed on the mathjax sponsors page .
that was a fab. explanation: can anyone pl. tell me what if there is any conclusion that can be derived from curl and divergence values. . say if a field has zero curl and non - zero divergence. . (or any other combo: both 0, both not 0 etc .) can we infer something ?
the corresponding curl would then have units (length ^ 2 / time) / (length ^ 2). that’s probably the meaning in the whirlpool example :) .
in figure 2, the water wheel rotates in the clockwise direction. hence, the z - component of the curl for the vector field in figure 1 is negative .
if the field lines of a vector field are rotating, the curl of that vector field gives the direction of axis of rotation about which the field lines are rotating .
there' s one simple answer, which i' m very surprised hasn' t been mentioned. use the book div, grad, curl, and all that .
so i went through (a) and showed that curl (v x f) = 2v. (doing cross products twice, which comes out pretty simple. )
here, $ \ nabla \ wedge \ psi $ is the gradient, and the second $ \ nabla \ wedge $ on the left plays the role of curl .
from there you can proceed by deducing properties of $ \ operatorname { curl } x $ sufficient to give its formula in coordinates and at the same time prove the theorem .
note: do not confuse div, grad, curl are dead with the above - recommended book div, grad, curl and all that: the former is modern and polemic, while the latter is traditional and tutorial. students who like either one generally will not like the other one; it is useful therefore to point students toward both books .
i loved learning about how differential forms and the exterior derivative generalize 3 - d vector calculus (div, grad, curl). differential forms are so elegant in comparison, work in arbitrary dimensions, and give rise to beautiful mathematics (e. g. de rham cohomology, hodge theory). and of course, the generalized stoke' s theorem is one of the prettiest equations: $ \ int _ { \ partial r } \ phi = \ int _ r d \ phi $ .
well look here for the definition of curl. if that' s what you' re looking for, you might want to find a copy of schey' s div, grad, curl, and all that - - he does of good job of explaining the limit - integral definitions. btw, you can also define the exterior derivative this way .
i am aware of an intuitive explanation for $ \ operatorname { curl } \ operatorname { grad } f = 0 $ (a block placed on a mountainous frictionless surface will slide to lower ground without spinning), and was wondering if there were a similar explanation for $ \ operatorname { div } \ operatorname { curl } f = 0 $ .
on the other hand, here the divergence and curl both take on the meaning of $ \ nabla \ wedge $. this is why differential forms likes to call them all aspects of $ \ nabla \ wedge $. but both of these identities for divergence, curl, and gradient have exactly the same expression in both differential forms and geometric calculus .
to be technical, curl is a vector, which means it has a both a magnitude and a direction. the magnitude is simply the amount of twisting force at a point .
try div, grad, curl and all that by schey. his first definition of curl is slightly different than mine (he defines it implicitly), but then on page $ 111 $ of the $ 3 $ rd edition he gives this as an
alternate definition
. his notation is a little different, but it is the exact same definition .
or intuitively, the curl is perpendicular to the divergence, which is an abuse of notation because divergence is really an operator, so this is as far as the analogy goes .
when i was younger, i envisioned curl (grad f) = 0 as the mathematical version of the statement
you can' t go uphill both ways to school
.
one way to approach the idea of the curl is through stokes' theorem, which says the circulation of vector field around a surface is equal to the flux of the curl across the surface: $ $ \ int _ { \ partial s } { \ bf f } \ cdot dr = \ iint _ s { \ rm curl } \; { \ bf f } \ cdot { \ bf n } \; ds $ $ where $ { \ bf n } $ is the surface normal. by the mean value theorem $ $ \ iint _ s { \ rm curl } \; { \ bf f } \ cdot { \ bf n } \; ds = { \ rm curl } \; { \ bf f } \ cdot { \ bf n } \ iint _ s ds = ({ \ rm curl } \; { \ bf f } \ cdot { \ bf n }) \; a (s) $ $ where $ { \ rm curl } \; { \ bf f } \ cdot { \ bf n } $ is evaluated at some point on $ s $. and so $ $ { \ rm curl } \; { \ bf f } \ cdot { \ bf n } = { \ hbox { circulation } \ over a (s) } $ $ if $ s $ is a small square normal to $ { \ bf n } $, the circulation will be maximized when $ { \ bf n } $ points in the direction of $ { \ rm curl } \; { \ bf f } $ .
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this macroscopic circulation of fluid around circles (i. e. , the rotation you can easily view in the above graph) actually is not what curl measures. but, it turns out that this vector field also has curl, which we might think of as “microscopic circulation. ” to test for curl, imagine that you immerse a small sphere into the fluid flow, and you fix the center of the sphere at some point so that the sphere cannot follow the fluid around. 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 rotation of the sphere measures the curl of the vector field $ \ dlvf $ at the point in the center of the sphere. (the sphere should actually be really really small, because, remember, the curl is microscopic circulation. )
thanks a lot. . now my concept of curl are a million times better. . but still i couold not uunderstand why exactly the curl of a non conoservative field cannot be zero. . i mean, conservative vector field need not be symmetrical. mand any vector field that is unsymmetrical can actually cause the circulation? please clearify this point. thanks a lot swetha
ok physical significance i understood. but i cannot relatate the output of nabla cross (vector field) with the physical urltoken is it relate with circulation / flow for curl / divergence .
- url ,
there are versions for unix, linux, windows, mac and other operating systems. for more information, visit http: / / curl. haxx. se .
we present a generalization of the div - curl lemma to a banach space framework which is not included in the almost existing generalizations. an example is shown where this generalization is needed .
@ msegade thank you for the question! i' ve been similarly curious about the curl ever since i took multivariable calculus. if you ever get to where you can read james clerk maxwell' s papers on the development of his equations for the electric and magnetic fields, you see him struggle to imagine just how the fields work and then the formulas for the curl just fall out - - it was mind - blowing. that guy' s imagination was something else. you feel that the curl is a real thing and not just some disembodied formula.
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 direction of the curl is the axis of rotation , as determined by the right-hand rule , and the magnitude of the curl is the magnitude of rotation . if the vector field represents the flow velocity of a moving fluid , then the curl is the circulation density of the fluid . a vector field whose curl is zero is called irrotational . the curl is a form of differentiation for vector fields . the corresponding form of the fundamental theorem of calculus is stokes ' theorem , which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve . the alternative terminology rotor , rotation or rotational and alternative notations rot f and ∇ × f are often used ( the former especially in many european countries , the latter , using the del ( or nabla ) operator and the cross product , is more used in other countries ) for curl f. unlike the gradient and divergence , curl does not generalize as simply to other dimensions ; some generalizations are possible , but only in three dimensions is the geometrically defined curl of a vector field again a vector field . this is a similar phenomenon as in the 3 dimensional cross product , and the connection is reflected in the notation ∇ × for the curl . the name " curl " was first suggested by james clerk maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by james maccullagh in 1839 . 
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 direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. if the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. a vector field whose curl is zero is called irrotational. the curl is a form of differentiation for vector fields. the corresponding form of the fundamental theorem of calculus is stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. the alternative terminology rotor, rotation or rotational and alternative notations rot f and ∇ × f are often used (the former especially in many european countries, the latter, using the del (or nabla) operator and the cross product, is more used in other countries) for curl f. unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. this is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl. the name " curl " was first suggested by james clerk maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by james maccullagh in 1839.
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 direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation. if the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. a vector field whose curl is zero is called irrotational. the curl is a form of differentiation for vector fields. the corresponding form of the fundamental theorem of calculus is stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. the alternative terminology rotor, rotation or rotational and alternative notations rot f and ∇ × f are often used (the former especially in many european countries, the latter, using the del (or nabla) operator and the cross product, is more used in other countries) for curl f. unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. this is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl. the name " curl " was first suggested by james clerk maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by james maccullagh in 1839.