in vector calculus , the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional euclidean space .

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 .

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 .
convert gradient _ dst . png gradient _ src . png \ - compose mathematics - define compose : args = ' 1 , 0 , 0 , 0 ' - composite \ mathematics _ multiply . png
strang points out in his cse book that div curl = 0 is the transpose of curl grad = 0 .
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convert gradient _ dst . png gradient _ src . png \ - compose mathematics - define compose : args = ' - 1 , 1 , 1 , 0 ' - composite \ mathematics _ screen . png
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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 .
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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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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 .
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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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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 .
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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
composite
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 . .
( formerly math 431 ) introduction to ordinary differential equations . first and second order linear differential equations , systems of linear differential equations , laplace transform , numerical methods , applications . ( this course is considered upper division with respect to the requirements for the major and minor in mathematics . )
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 .
various topics that might enrich an elementary school mathematics program , including probability and statistics , the integers , rational and real numbers , clock arithmetic , diophantine equations , geometry and transformations , the metric system , relations and functions . for pre - early childhood and pre - elementary education majors only .
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 .
this is a rigorous introduction to some topics in mathematics that underlie areas in computer science and computer engineering , including graphs and trees , spanning trees , and matchings ; the pigeonhole principle , induction and recursion , and generating functions . the course integrates learning mathematical theories with applications to concrete problems from other disciplines using discrete modeling techniques . student groups will be formed to investigate a concept or an application related to discrete mathematics , and each group will report its findings to the class in a final presentation . this course satisfies the university ' s integrative experience ( ie ) requirement for math majors .
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 .
an individual project of guided reading and research culminating in a substantial paper or other piece of work which can be meaningfully evaluated to assign a letter grade ; may not be taken on a pa / fl basis . students engaged in preparation of a senior thesis ordinarily should take applied mathematics 99r instead .
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 } $ .
this course is a basic introduction to combinatorics and graph theory for advanced undergraduates in computer science , mathematics , engineering and science . topics covered include : elements of graph theory ; euler and hamiltonian circuits ; graph coloring ; matching ; basic counting methods ; generating functions ; recurrences ; inclusion - exclusion ; and polya ' s theory of counting .
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 .


player tracking is a new technology being used in the nba to increase basketball statistical data collection and enhance advanced metrics .

it began use in the nba at the start of the 2013 – 14 season , and it is now in use in all 29 arenas , following trials during the previous season in 15 arenas .

usa today said that " for fans who want to understand the game on a deeper level , this is a huge development " .

player tracking ( national basketball association )

exploring game performance in the national basketball association using player tracking data . - pubmed - ncbi
deitch jr , starkey c , walters sl , moseley jb . injury risk in professional basketball players : a comparison of women ' s national basketball association and national basketball association athletes .
simenz cj , dugan ca , ebben wp . strength and conditioning practices of national basketball association strength and conditioning coaches . journal of strength and conditioning research / national strength & conditioning association . 2005 ; 19 ( 3 ) : 495–504 .
citation : sampaio j , mcgarry t , calleja - gonzález j , jiménez sáiz s , schelling i del alcázar x , balciunas m ( 2015 ) exploring game performance in the national basketball association using player tracking data . plos one 10 ( 7 ) : e0132894 . urltoken
the nba began using player - tracking technology in 2009 and became the first u . s . professional sports league to use player tracking for every game in 2013 .
new york and london , sept . 22 , 2016 – the national basketball association ( nba ) today announced a multiyear partnership with sportradar and second spectrum that will expand the league’s global distribution of statistics and will implement a new , state - of - the - art leaguewide player - tracking system .
messersmith ll , corey sm . the distance traversed by a basketball player . research quarterly american physical education association . 1931 ; 2 ( 2 ) : 57–60 .
the national basketball association has been using sportvu since 2009 to gather data on players through cameras in roof beams . the data collected is unconventional and not found on box scores , such as how many times a player touches the ball per shot .
at the nba level , wearable technology can only be used in practice for now . in - game use must be collectively bargained with the national basketball players’ association , and executive director michele roberts can see the value .
law360 , new york ( september 22 , 2016 , 8 : 19 pm edt ) - - the national basketball association has partnered with sports data firms sportradar and second spectrum in a deal that the league said will help it get statistics and game feeds to international gaming operators and improve its advanced player tracking .
like when harvard students were toying with home computers in the early 1980s , there’s no telling where this player tracking technology might take basketball analytics .
player distributions in the seven clusters contrasted against player category , nba first team and player specific court position .
mangine gt , hoffman jr , wells aj , gonzalez am , rogowski jp , townsend jr , et al . visual tracking speed is related to basketball - specific measures of performance in nba players . journal of strength and conditioning research / national strength & conditioning association . 2014 ; 28 ( 9 ) : 2406–14 .
now the national basketball association is finally trying to catch up with a more data - friendly approach . the information reported yesterday that the nba has partnered with stats llb last year to set up a six - camera player - tracking system in all 30 team arenas . it can monitor the ball movement and all the players on the court 25 times a second , producing massive amounts of data .
complex systems and sport research group , national institute of physical education of catalonia ( inefc ) , barcelona , spain .
this distinction between quantifiable and non - quantifiable player skills may be on the verge of disappearing . beginning in the 2013–2014 season , all thirty arenas of the national basketball association ( nba ) contain a system of cameras and tracking software that allow the spatial coordinates of all players and the ball to be recorded and processed digitally [ 1 , 2 ] . in this way essentially every aspect of the game is made accessible for quantitative analysis—every pass , screen , and defensive rotation can in principle be analyzed quantitatively by those with access to this “player tracking” data .
the national football league placed tracking sensors in the footballs used for the 2016 preseason as well as thursday night games . sensors and receivers throughout nfl stadiums , footballs , and player gear track location as well as throwing speeds , velocity , and acceleration .
gonzalez am , hoffman jr , rogowski jp , burgos w , manalo e , weise k , et al . performance changes in nba basketball players vary in starters vs . nonstarters over a competitive season . journal of strength and conditioning research / national strength & conditioning association . 2013 ; 27 ( 3 ) : 611–5 .
schelling x , calleja - gonzalez j , torres - ronda l , terrados n . using testosterone and cortisol as biomarker for training individualization in elite basketball : a 4 - year follow - up study . journal of strength and conditioning research / national strength & conditioning association . 2015 ; 29 ( 2 ) : 368–78 .
“nnba partners with stats llc for tracking technology” . nba official release . september , 2013 . urltoken
the nba began using stats sportvu in 2009 and in 2013 became the first u . s . professional sports league to use player tracking for every game .
fans and media around the world can continue to access comprehensive league data and statistical information , along with advanced player - tracking data at nba . com .
currently , there are over 90 new player tracking statistics , and data for all 482 nba players from the 2013 - 2014 regular season are available . separate records exist for players who played for different teams throughout the season , including a record for overall performance across all teams . brief descriptions of the newly available statistics adapted from the nba player tracking statistics website ( player tracking , 2016 ) are provided for reference and will be helpful in better understanding the analysis going forward .
“wearable technology is an important part of the player health conversation , ” said mike bass , the nba’s executive vice president of communications . “we’ll work with the players’ association to establish the appropriate operational and data privacy rules . ”
major league baseball , for example , has turned into a data hotbed with the creation of
moneyball ,
a team management concept that heavily relies on statistical analysis . the national football league , too , has recently adopted more data tracking technology , such as rfid , to monitor player movement and distance run in real - time .
player tracking data provides a platform for the creation of new basketball statistics that can dramatically improve the ability to evaluate and compare player performance . however , the increasing size of this new data source presents challenges in how to efficiently analyze the data and interpret findings . a scalable analytical framework is needed that can effectively reduce the dimensionality of the data while retaining the ability to compare player performance .
the nfl has invested in sports data collection and analysis by placing tracking sensors in footballs . photo credit : the atlantic
with constant movement , a controlled environment and only 11 moving parts , nba basketball was perfect .
to better understand the additional value of the player tracking data , pc scores and sdi can be recalculated using only traditional statistics . this approach identifies demar derozan as the most similar player to tony parker although derozan and parker have an sdi of 75 using player tracking statistics ( 89 other players have a lower sdi compared to tony parker ) . to better explore the difference between the comparisons using traditional and player tracking statistics , derozan is included in the next set of comparisons along with j . j . barea and brandon jennings from table 6 . see fig . 6 for a comparison of the pc scores and table 7 for selected statistics for these players .
piette j , anand s , pham l . evaluating basketball player performance via statistical network modeling . in : mit sloan sports analytics conference ; 2011 . available from : urltoken
haglund david , 2012 . “does basketball actually have 13 positions ? ” slate . may 2012 . urltoken
shambaugh jp , klein a , herbert jh . structural measures as predictors of injury in basketball players .
physiologically , a professional basketball game is quite demanding . 1 in one 48 - minute game , a single player may travel a distance 1 greater than 6000 m via a variety of short , moderate - to high - intensity exertions 2 that include sprints and quick changes of direction as well as powerful jumps . over the course of an entire national basketball association ( nba ) season , a player may participate in 82 regular - season competitions ( 2–5 games per week ) , in addition to daily practice sessions and preseason and postseason practices and games . although the total demand on a single player may vary as a result of a player ' s specific roles ( eg , position responsibilities ) , playing time , travel schedule , and days off , 3 the ability to remain healthy during such a demanding schedule may depend on underlying physical factors that may make some athletes more susceptible to injury .
basketball has never been the most statistically advanced game , at least compared to its counterparts baseball and football .
chicago and new york – the nba and stats llc today announced an expansion of their long - standing partnership that will provide media outlets around the world with an opportunity to use much of the same stats sportvu player tracking data used by nba teams for scouting , player development and training .
the sportvu cameras track every player and the basketball 25 times a second . the data the cameras collect can tell how fast a player moves , how close he was defended on a shot , how many times he dribbled and a plethora of other fascinating nuggets .
throughout the year , nba tv , urltoken and the nba app will continue to use stats sportvu player tracking data to deliver the most comprehensive statistical breakdown of every game . the nba and stats originally partnered in 2006 .
this work could be extended by incorporating new data from the 2014 - 2015 season to see if principal component dimensions and player principal component scores change significantly from one season to another . this would also provide additional data for players who didn’t see much playing time in the 2013 - 2014 season . additionally , player tracking data at the game level could greatly extend this analysis by tracking how players’ statistical profiles change throughout the course of the season .
in closing , we look forward to future work that can apply the approach we have advocated here to a large corpus of player tracking data . such work will likely spur the development of new network - style models and refined statistical inference algorithms , but we remain confident that the general strategy we have described can be successful if given sufficient attention and the full power of many games worth of player tracking data . if this is indeed the case , then perhaps the network - based description of basketball will provide a suitably revolutionary application for a revolutionary new data source .
the national basketball association ( nba ) from the united states of america is the most competitive basketball league in the world , with a competition period in regular season comprising 82 games spanning approximately 24 weeks . the coaching staff must prepare and oversee the training loads on players throughout the entirety of the competition period , a complex process that places a great amount of physiological stress on the athletes [ 1 ] . this process also requires managing the significant differences in work demands introduced by position - specific game behaviors and player status ( e . g . , starting vs non - starting players ) , as well as adjusting throughout the season to several changing unpredictable constraints such as player injuries . thus , the ongoing planning and monitoring of practice sessions and game performance is critical for optimizing the decisions on individual training loads taken by coaching staff .
second spectrum will begin installing its player - tracking system in all nba arenas over the course of the upcoming season so that it will be fully functional by the 2017 - 18 season , when the company becomes the official optical tracking provider of the nba . second spectrum will provide advanced statistics , such as speed , distance , drives , paint touches and defensive impact , based on its unique technology for machine understanding of sports for comprehensive analysis of players and teams . additionally , second spectrum will continue to work individually with teams and league partners on providing custom solutions and products based on player - tracking data .
mcinnes se , carlson js , jones cj , mckenna mj . the physiological load imposed on basketball players during competition .
player pc scores for first four components for tony parker comparison ( color online ) .
player pc scores for first four components for anthony morrow comparison ( color online ) .
mckay gd , goldie p , payne wr , oakes b . ankle injuries in basketball : injury rate and risk factors .
lowe z . seven ways the nba’s new camera system can change the future of basketball ; 2013 . available from : urltoken
this article explores the utility of the newly available nba player tracking data in evaluating and comparing player and team abilities and playing styles through their statistical profiles . using pca , we identify and interpret four principal components that capture over 68 % of the variance in the nba player tracking data set and compare players along these new dimensions . a simple measure of comparison between two players or teams based on principal component scores , sdi , is also introduced . this framework is scalable as it can incorporate existing and new statistics that will emerge to reconstruct the principal component dimensions and sdi for improved comparisons .
alagappan muthu , 2012 . “from 5 to 13 . redefining the positions in basketball” . 2012 mit sloan sports analytics conference . urltoken
a sample of player data for wins and losses catapult sports collects for its nba clients .
in sec . 4 we show how player tracking data can be used to learn the set of all player skills , so that the effectiveness f of the offense can be evaluated for any set of play frequencies p ≡ { p α } and any combination of players . but first we outline a simple model of a basketball offense , which we call the “high / low” model , that will serve as a proof of concept for the network approach advocated in this paper .
that ' s just the tip of the iceberg . the sportvu tracking numbers allow fans to go deep on defensive impact , rebounding and time of possession for players .
video footage from the entire court was unavailable making assessment of the nba tracking data impossible . the nba non - tracking data ( e . g . , assists , steals or defensive rebounds ) however was assessed for reliability as follows . two games were selected at random and analyzed conjointly through systematic observation by two experts . the minimum cohen’s κ value for all variables exceeded 0 . 91 demonstrating high inter - rater reliability [ 26 ] between the nba non - tracking data and the two experts .
maheswaran r , chang y - h , henehan a , danesis s . deconstructing the rebound with optical tracking data . mit sloan sports analytics conference 2012 . 2012 .
when a player is within 5 feet of the basket and within 5 feet of the offensive shooter , opponents’ shooting statistics are available to measure how well a player can protect the basket .
} also involves the quantitative definition of all player skills that are relevant to the offense . one specific definition of player skills and success rates is discussed in sec . 2 and diagrammed in
those cameras are part of a system called sportvu , and it has the potential to change everything we know about analyzing nba basketball .
shirley k . a markov model for basketball . in : new england symposium for statistics in sports ; 2007 . boston , ma .
the answer seems simple : the players’ union , which is far stronger in the national basketball association than in comparable leagues in other countries . through the collective - bargaining process , the union has carved out concessions that are keeping so - called “wearables” off the court for the time being , and keeping the data out of the hands of coaches and managers . that’s because , as much as teams try to assure players that the technology will only be used to help them—and even though many players , including stars , have taken to using them in their personal training regimens—the players know the data very well could end up costing them millions when they sign their next contracts . but all that could change in the next few years , not just in basketball but in other leagues as well .
archival data were obtained from open - access official nba records for 1230 games played during the 2013–2014 regular season ( available at urltoken , these records contained both non - tracking and tracking data ) . a total of 30 teams played 82 games between october 29 , 2013 and april 16 , 2014 . the gathered database had records of game performances from 548 players . the cases of player transfer between teams were counted as two different records .
the power of this network description becomes apparent when one imagines coupling it with the full player tracking data set . indeed , player tracking data allows one to reconstruct directly how the offense progresses from the inbounds pass to a made basket . the success rate of each step along the way provides information about the effectiveness of links in the network , which in turn are a reflection of the skill levels of the players involved . thus , by “watching” the performance of the offense with player tracking data , it should be possible to learn the skill levels of the players involved , including any skills that affect the team’s offensive performance . once these skills are known , one can predict quantitatively how the offense will perform when called upon to run different plays or to substitute different players whose skills are also known .
if you want to see just how much of derrick rose ' s explosiveness has returned , the tracking service offers a section for drives to the basket . rose has made 11 drives over two games and scored four points . those numbers will look different once rose becomes acclimated to playing nba - level basketball again .
within this model , we further assume that the success rate of any given link depends only on the skills of the player holding the ball . specifically , the success rate of a pass by player
the slippery slope begins with playing time being limited , even if it’s for the player’s own good .
maheswaran r , chang yh , henehan a , danesis s . deconstructing the rebound with optical tracking data . in : mit sloan sports analytics conference ; 2012 . available from : urltoken
weil s . the importance of being open : what optical tracking data can say about nba field goal shooting . in : mit sloan sports analytics conference ; 2011 . available from : urltoken
oliver dean , et al . 2007 . “a starting point for analyzing basketball statistics . ” journal of quantitative analysis in sports 3 . 1 web .
nba ' s new tracking stats give fans a window to the future data from the league ' s new sportvu cameras is now available to the public . check out this story on urltoken urltoken
the potential benefits to coaches and observers is apparent , simply by taking a look at what the system can tell us about the basic areas of basketball .
plisky pj , rauh mj , kaminski tw , underwood fb . star excursion balance test as a predictor of lower extremity injury in high school basketball players .
noted previously , the variables obtained from the tracking systems allow use of court locations for better understanding several game statistics . therefore , these results increase knowledge of basketball game behavior by identifying key performance variables and by reducing prior emphasis on the importance of distance covered and velocity . the reclassification obtained was very high and hence affirms accuracy of the mathematical model .
in leagues around the world , wearable biometric trackers that provide these statistics are the norm ; in some countries , they can even display a player’s heart rate in real time on the jumbotron or home broadcast . so why isn’t this a common sight in american professional basketball ?
to examine the relationship among skeletal muscle architecture , lower body power , and games missed because of lower extremity injury ( % miss ) in professional basketball players .
schiltz m , lehance c , maquet d , bury t , crielaard j - m , croisier j - l . explosive strength imbalances in professional basketball players .
maymin p . acceleration in the nba : towards an algorithmic taxonomy of basketball plays . in : mit sloan sports analytics conference ; 2013 . available from : urltoken
the nba released its sportvu tracking technology and statistics to the public on friday , and while it ' s hard to see the results with so few games played , we know what will be available .
muscle architecture and power were assessed in basketball players from the orlando magic professional basketball organization during the week immediately before the beginning of the 2012–2013 regular season . athletes ' bilateral differences in both muscle architecture and power were compared with games missed because of lower extremity injury over the course of the regular season ( 82 games ) .
only players who played at least half the 2013 - 2014 regular season , 41 games , are included in this analysis . this restriction is made to reduce the influence of player statistics derived from only a few games played . also fields containing season total statistics and per game statistics are dropped from the analysis since they could be influenced by number of games and minutes played throughout the season . instead , per 48 minutes , per touch , and per shot statistics are used . the final data set contains 360 player records each containing 66 different player tracking statistics .
however , the high dimensionality of this new data source poses major challenges as it demands more computational resources and reduces the ability to easily analyze and interpret findings . we must find a way to reduce the dimensionality of the data set while retaining the ability to compare player performance . one method that is particularly well - suited for this application is principal component analysis ( pca ) which identifies the dimensions of the data containing the maximum variance in the data set . this article applies pca to the nba player tracking data to discover four principal components that account for 68 % of the variability in the data for the 2013 - 2014 regular season . these components are explored in detail by examining the player tracking statistics that influence them the most and where players and teams fall along these new dimensions .
esteves pt , de oliveira rf , araujo d . posture - related affordances guide attacks in basketball . psychol sport exerc . 2011 ; 12 ( 6 ) : 639–44 .
kerr , ringleader of the most entertaining basketball show on earth this season , didn’t take the decision lightly . he even emailed fans to explain the reasons behind his decision .
* the nba - wide shooting percentage is significantly higher when the shooter doesn’t take any dribbles . this confirms what any basketball observer suspected : ball movement equals offensive success .
manzi v , d ' ottavio s , impellizzeri fm , chaouachi a , chamari k , castagna c . profile of weekly training load in elite male professional basketball players .
we at squadfusion have also written on the site before about how underdogs in football and basketball can strategize with data by finding what is unconventional and outwitting their stronger opponent .
for models that do not make explicit consideration of such effects , one should be careful when making predictions about hypothetical lineups to ensure that each player’s usage rate in the new offense is not significantly different than the player’s usage rate in the data from which their skills are inferred . otherwise , one can get spurious predictions based on unrealistically robust estimates of player skills .
with numerous player tracking statistics already available and the potential to develop infinitely many more , it is difficult to extract meaningful and intuitive insights on player comparisons . now that the data are available for more granular and detailed comparisons , a methodology is needed that can analyze the entirety of the data set to extract a handful of dimensions for comparisons . these dimensions should be constructed in a way that ensures optimality in differentiating players ( i . e . dimensions should retain the maximum amount of player separability possible from the original data ) and can be understood in terms of the original statistics .
this study aimed to compare game performances of all - star and non all - star basketball players and to identify and describe different basketball game performance profiles in the nba . in general terms , key performance indicators were identified that discriminate all - star players from non all - stars and , also , the different groupings of performance profiles in competition .
malarranha j , figueira b , leite n , sampaio j . dynamic modeling of performance in basketball . international journal of performance analysis in sport . 2013 ; 13 : 377–86 .
in this paper , principal component analysis ( pca ) is used to identify four components accounting for 68 % of the variation in player tracking data from the 2013 - 2014 regular season . the most influential statistics on these new dimensions are used to construct intuitive , practical interpretations . in this high variance , low dimensional space , comparisons across any or all of the principal components are possible to evaluate characteristics that make players and teams similar or unique . a simple measure of similarity between player or team statistical profiles based on the four principal components is also constructed . the statistical diversity index ( sdi ) allows for quick and intuitive comparisons using the entirety of the player tracking data . as new statistics emerge , this framework is scalable as it can incorporate existing and new data sources by reconstructing principal component dimensions and sdi for improved comparisons .
in this paper , we suggest a method for achieving these goals using player tracking data . we first outline generally how to build a network model of the offense that relates player skills to the effectiveness of a given play , and we provide a specific , simplified example of such a model . we then discuss the process of learning player skills from player tracking data , and we show that in general these skills cannot be measured directly from the data due to a lack of information about which play a team is attempting at the moment of a failed offensive sequence . nonetheless , we show that this issue can be addressed using an iterative procedure based on bayesian inference . the effectiveness of this procedure is then demonstrated by means of a test on simulated data . finally , we use our simplified network model to examine hand - recorded data from the 2011 playoff series between the oklahoma city thunder and the memphis grizzlies . we show that , despite a very limited data set , we are able to correctly predict a player’s interactions with a particular lineup based on data from his performance with a different lineup .
karipidis a , fotinakis p , taxildaris k , fatouros j . factors characterizing a successful performance in basketball . j hum movement stud . 2001 ; 41 ( 5 ) : 385–97 .
sampaio j , gonçalves b , rentero l , abrantes c , leite n . exploring how basketball players ' tactical performances can be affected by activity workload . sci sport . 2014 .
as the official player tracking partner of the nba , stats sportvu data will now be used by espn , nba on tnt , and bleacher report . the data will also be made available to other media outlets around the globe to provide fans with more insight into lineups , substitutions , and play calling , as well as analysis on player and team performance including shooting , passing , and rebounding . previously , the nba’s sportvu data was only available to nba teams and through nba digital assets .
additionally , a simple measure of similarity in statistical profiles between players and teams based on the principal components is proposed . the statistical diversity index ( sdi ) can be calculated for any pairwise player combination and provides a fast and intuitive method for finding players with similar statistical performances along any or all of the principal component dimensions . this approach is also advantageous from the standpoint of scalability . the possibilities to derive new statistics from the player tracking data are endless , so as new statistics emerge , this approach can again be applied using the new and existing data to reconstruct the principal components and sdi for improved player evaluation and comparisons .
professional basketball players have demanding schedules that , in combination with certain underlying physical characteristics and side - to - side strength and power imbalances , may make them vulnerable to lower extremity injuries .
lombardo s , sethi pm , starkey c . intercondylar notch stenosis is not a risk factor for anterior cruciate ligament tears in professional male basketball players : an 11 - year prospective study .
professional sports leagues such as the nba and nfl place sensors and receivers across their stadiums and player gear . photo credit : the atlantic
teams can break down each player ' s efficiency into more detailed sub - categories , like touches per shot or points per touch .
here ' s what an in - game analysis would look like . you can track the distance run or speed for each player .
remmert h . analysis of group - tactical offensive behavior in elite basketball on the basis of a process orientated model . eur j sport sci . 2003 ; 3 ( 3 ) : 1–12 .
this article is organized as follows . section 2 describes the player tracking data and data processing . section 3 provides the analysis and interpretations for the four principal components in detail , showing how players and teams can be compared across these new dimensions . section 4 introduces the calculation for sdi and two case studies where principal component scores and sdi are used to find players with similar statistical profiles to all - star tony parker and role player anthony morrow for personnel management purposes . 5 conclusion concludes the article with final remarks .
unlike in the “random agents” model detailed in appendix a , for the “coached agents” model that is the main subject of this paper , one cannot , in general , measure player skills directly from player tracking data . this is because player tracking data provides only a sequence of plays as they are performed by the players , and does not give any information about the team’s original intentions ( i . e . , the play as diagrammed by the coach ) . consider , for example , the problem of determining a player’s passing skill in the high / low model . a typical possession within this model might progress from the state 1h to the state 3h to a turnover ( denoted , for brevity , 1h → 3h → t ) . but in this sequence it is not clear how the turnover by the small forward should be interpreted . was the turnover the result of a failed high pass ? a failed low pass ? a failed drive ? in order to determine how to assign blame for this turnover among the player’s various skills , one needs to know something about which plays the team tends to attempt . thus , the frequencies of play calls { p α } must be inferred simultaneously with the player skills . in this endeavor one can employ a statistical inference algorithm , such as the one we outline below .
numerous applications in personnel management exist for the use of sdi and principal component scores in evaluating and comparing player and team statistical performances . two specific case studies are presented to show how these tools can be used to quickly identify players with similar statistical profiles to a certain player of interest for the purpose of identifying less expensive , similarly skilled players or finding suitable replacement options for a key role player within the organization .
“opinions are being formed about a player’s potential based on how some people are reading , reviewing and assessing the data , ” roberts said .
aglioti sm , cesari p , romani m , urgesi c . action anticipation and motor resonance in elite basketball players . nat neurosci . 2008 ; 11 ( 9 ) : 1109–16 . pmid : 19160510
apostolidis n , nassis gp , bolatoglou t , geladas nd . physiological and technical characteristics of elite young basketball players . j sport med phys fit . 2004 ; 44 ( 2 ) : 157–63 .
another way the pc scores can be used to compare player statistical profiles is to combine them to produce one measure of how different one player’s statistical profile is from another . here a simple calculation based on the sum of squared difference between the two players’ four pc scores is proposed .
the good that data can bring to sports includes better team strategy , player performance , and medical care . the trackers and cameras used by the nfl and nba are also used by numerous other soccer , basketball , college football leagues . the information technology brings is transforming our understanding of athletics and will only continue to spread to every field .
in sec . 4 we showed that our statistical inference algorithm can correctly infer the skill levels of players in a simulated offense . given this success , the prospect of using the algorithm to learn the traits of actual nba players is naturally enticing . a successful inference scheme , coupled with a suitably sophisticated network model , would allow one to learn the skill levels of nba players from player tracking data and thus predict how the team’s effectiveness would be affected by any number of changes , including player trades , substitutions , or altered play calling .
} of each player . thus , one can completely determine the expected number of points scored by the team for a given use of plays {
engelmann j . new player evaluation technique for players of the nba . in : mit sloan sports analytics conference ; 2011 . available from : urltoken
on friday , the nba launched the public database based on tracking technology from stats llc ' s innovative sportvu cameras . while there are limitations to what ' s available to fans , the release is certainly an upgrade and another level of insight into the league .
ben abdelkrim n , castagna c , jabri i , battikh t , el fazaa s , el ati j . activity profile and physiological requirements of junior elite basketball players in relation to aerobic - anaerobic fitness .
gaida j , cook j , bass s , austen s , kiss z . are unilateral and bilateral patellar tendinopathy distinguished by differences in anthropometry , body composition , or muscle strength in elite female basketball players ?
numerical evaluation of basketball players has long been based on box score statistics . such evaluations , by nature of the limited data set from which they draw , generally center around a player’s contributions to the five positive statistics—points , rebounds , assists , steals , and blocks—while neglecting more nuanced aspects of the player’s value , such as his / her ability to make high quality ( non - assist ) passes , or set good screens , or rotate effectively on defense . these less easily quantifiable aspects of a player’s performance are traditionally evaluated only qualitatively , informed by the intuition of a coach or analyst who has spent a significant amount of time watching the players perform .
derozan and parker have similar point totals per 48 minutes , but parker generates more driving points compared to catch and shoot points while derozan is more balanced . additionally , derozan doesn’t match the others in terms of assist categories , touches , and passes . this helps explain why derozan’s pc 2 score is much smaller than the others as catch and shoot offense and lack of passing and assists contribute to negative pc 2 scores . shot type , touches , and passes are key aspects of the player tracking statistics that add value by improving player comparisons beyond simple number of points and assists . with player tracking statistics , j . j . barea rises as a less expensive option to parker’s $ 12 . 5m salary who had the most similar statistical performance to parker in the 2013 - 2014 season as measured by sdi . this can be seen in the similarities among pc scores and the selected statistics and shows how sdi can provide a quick method for identifying similar players for further detailed comparisons .
the mean squared error in the inferred value of player skills for a simulated high / low offense as a function of the number of possessions analyzed .
sportvu tracks every player movement , every pass , every shot , every touch - - everything . at this point , it tracks more information than teams know what to do with . every executive interviewed for this article agreed they weren’t even using 10 percent of the information this system could provide . and they all agreed this is the future of advanced basketball analytics .
kozar b , vaughn re , whitfield ke , lord rh , dye b . importance of free - throws at various stages of basketball games . percept motor skill . 1994 ; 78 ( 1 ) : 243–8 .
in sports more than in any other professions , the boundary between the two has always been blurry . with few exceptions , it’s very rare for somebody who’s not at peak physical fitness to reach the upper echelons of professional sports , especially in basketball , where only a handful of players under six feet tall have ever truly thrived . some of the data the leagues already collect could arguably be construed as falling in either category : when the nba tracks how far a player runs in an average game , or a baseball team uses special slow - motion cameras to evaluate the angle of a player’s elbow as he releases a pitch , is that dissecting that player’s labor or studying his body ?
buchheit m , allen a , poon tk , modonutti m , gregson w , di salvo v . integrating different tracking systems in football : multiple camera semi - automatic system , local position measurement and gps technologies . j sport sci . 2014 ; 32 ( 20 ) : 1844–57 .
theoretically , a pre - draft mri could raise a player’s value : “if an mri is clean , ” says jeff passan , whose book the arm details the history of ucl tears and the surgeries that fix them , “what’s preventing the player from going back to the team and saying , ‘your doctor tells me i have a beautiful elbow , i want $ 300 , 000 more’ ? ” but there’s no way for a player to know until he gets his results , at which point the team’s leadership will always know better what to look for , giving them all the leverage they need to use whatever they find against the player .
of course , it is important to note that the apparent skill levels of the players , which define the success rates f ij of different attempted transitions , depend not only on the skills of the offensive players but on the quality of the defense being faced . in this sense any player skill is understood to be defined relative to a given defense . when considering the performance of a player over an entire nba season , for example , such skills are naturally defined relative to the league - average defense . one could also imagine considering a player’s performance against a specific opponent , and thus defining the player’s skills relative to that particular matchup .
principal component analysis ( pca ) , developed by karl pearson ( pearson , 1901 ) and later by hotelling ( hotelling , 1933 ) , is a particularly well - suited statistical tool that can accomplish this task through identifying uncorrelated linear combinations of player tracking statistics that contain maximum variance . interested readers can find a brief technical introduction to pca in appendix a . components of high variance help us better differentiate player performance in these directions in hopes that the majority of the variance will be contained in a small subset of components . this simple and intuitive approach to dimension reduction provides a platform for player comparisons across dimensions that best separate players by statistical performance and can be implemented withoutexpensive proprietary solutions , providing more visibility into how the method works at little to no additional cost .
franks a , miller a , bornn l , goldsberry k . characterizing the spatial structure of defensive skill in professional basketball . arxiv : 14050231 [ stat ] . 2014 may ; arxiv : 1405 . 0231 . available from : urltoken
sampaio j , janeira m . statistical analyses of basketball team performance : understanding teams’ wins and losses according to a different index of ball possessions . international journal of performance analysis in sport . 2003 ; 3 ( 1 ) : 40–9 .
perše m , kristan m , kovačič s , vučkovič g , perš j . a trajectory - based analysis of coordinated team activity in a basketball game . computer vision and image understanding . 2009 ; 113 ( 5 ) : 612–21 .
miller a , bornn l , adams r , goldsberry k . factorized point process intensities : a spatial analysis of professional basketball . arxiv : 14010942 [ stat ] . 2014 jan ; arxiv : 1401 . 0942 . available from : urltoken
of course , the real impact of the approach suggested here awaits its application to the actual player tracking data . under the influence of such data , the simple high / low model will likely be replaced by richer models , including perhaps models that incorporate a finer spatial grid or a continuous description of spatial positions . such detailed models may require one to adapt more advanced techniques from spatial statistics ( as , for example , in refs . [ 15 , 16 ] ) .
in summary , this study provided analysis of an nba regular season using player - tracking variables and notation data . it was found that all - star players performed consistently better than non all - star players within 12 feet of the basket , possibly a result of optimized attention processes that are key for perceiving the required appropriate environmental information for action production . in addition , different groupings were identified based on playing performance , particularly in relation to the roles of scoring , passing , defensive and all - round duties . these findings can be used to optimize preparation for individual player groupings and , ultimately , improve game performances of the players and teams .
tony parker dribbled 839 times in a 107 - 96 win over the thunder on feb . 4 , the most dribbles by a player in a sportvu game this season .
shows the resulting mean squared error in the inferred player skills as a function of the number of possessions analyzed . the results suggest that after a season’s worth of data the player skills are known to within a few percent , and that the inferred value of skills generally converge to the actual values after a few hundred possessions . this analysis uses
that is not captured directly by our network description . westbrook is generally called upon to carry a significantly larger offensive load than conley , a fact that is reflected in the much larger percentage of okc’s possessions that end with westbrook taking a shot or committing a turnover : about 35 % , as compared to only about 15 % for conley . in basketball analysis , it is generally accepted that larger usage rates for a given player are correlated with declining success rates , due primarily to the increased focus such a player receives from the defense . such a relationship is called a “skill curve” [
sampaio j , janeira m , ibanez s , lorenzo a . discriminant analysis of game - related statistics between basketball guards , forwards and centres in three professional leagues . european journal of sport science . 2006 ; 6 ( 3 ) : 173–8 .
using principal component scores and sdi , several use cases are presented for improved personnel management . team principal component scores are used to quickly profile and evaluate team performance , more specifically how new york’s lack of ball movement negatively impacted success despite high average scoring efficiency as a team . sdi is used to identify players across the nba with the most similar statistical performances to specific players . all - star tony parker and shooting specialist anthony morrow are used as two examples and presented with in - depth comparisons to similar players using principal component scores and player tracking statistics . this approach can be used in salary negotiations , free agency acquisitions and trades , role player replacement , and more .
unfortunately , player tracking data is not yet available in the public domain . thus , we devised the following basic test of our general strategy . in order to obtain some “poor man’s player tracking data” , we recorded by hand data corresponding to 401 possessions from the 2011 playoff series between the oklahoma city thunder ( okc ) and the memphis grizzlies . this data is described in more detail in the methods section . possessions were then sorted according to which lineup the team had on the floor . for example , the memphis starting lineup of conley - allen - young - randolph - gasol played 84 possessions together during the first three games , while the okc starting lineup of westbrook - sefolosha - durant - ibaka - perkins played 111 possessions ; we refer to these as “lineup 1” for each team . memphis’s second - most - used lineup was conley - mayo - battier - randolph - gasol , while for okc it was westbrook - harden - durant - collison - ibaka ; these we denote “lineup 2” .
ibanez sj , sampaio j , feu s , lorenzo a , gomez ma , ortega e . basketball game - related statistics that discriminate between teams ' season - long success . european journal of sport science . 2008 ; 8 ( 6 ) : 369–72 .
but it’s in basketball , which has already seen multiple dustups over wearables , and where the league has taken to actively pitching the devices at its annual tech summit , that the conversation’s been the most instructive when it comes to understanding where the controversy lies .
“to the extent that the team and the player could come up with a better advantage based on the information , who’s got a problem with that ? ” she told cbssports . com .
mckechnie’s observation was backed up by catapult’s work with a d - league team last season , yielding in - game data showing that only 15 . 5 percent of player movement was forward .
gomez ma , lorenzo a , ibanez sj , sampaio j . ball possession effectiveness in men ' s and women ' s elite basketball according to situational variables in different game periods . j sports sci . 2013 ; 31 ( 14 ) : 1578–87 . pmid : 23679867
close shots : any jump shot taken by a player on any touch that starts within 12 feet of the basket , excluding drives . gathered variables include close ppg or ppm and fg % .
just a sample of the twitter backlash directed at kerr : “nothing like spending $ 250 on tickets for my son’s 7th birthday only for you to bench his favorite player . thanks steve . ”
ben abdelkrim n , el fazaa s , el ati j . time - motion analysis and physiological data of elite under - 19 - year - old basketball players during competition . british journal of sports medicine . 2007 ; 41 ( 2 ) : 69–75 ; discussion pmid : 17138630
pca is most useful when the majority of the total variance across variables are captured by only a few of the principal components , thus the dimension reduction . if this is the case for the nba player tracking data set , it means that we are able to retain the ability to differentiate player performance without having to operate in such a high dimensional space . figure 1 scree shows the variance captured by each principal component . note the variance captured in the first principal component is very high and decreases drastically through the first four components . after that , the change in variance is relatively flat , forming an elbow shape in the plot . this means that the variances captured by the fifth component onward are very similar and much smaller than the first four components . moreover , the first four components capture 68 % of the variance across the original variables , so we utilize these four components going forward to analyze and compare player performance and playing styles .