vector spaces, leading to differential forms

intro to differential forms

i mentioned a while back that i wanted to make a thread about differential forms. differential forms are a vital aspect in any mathematical physicists toolbox. i had written a couple of pages of the thread during spring break, but i haven t had the time to do anything since then. unfortunately most of what i wrote was just general vector space primer, not differential forms.

still differential forms are fairly simple and i think anyone with a few college level maths courses can master them. i am going to post what i had written, rather than just forgetting it, and hopefully i can finish it up pretty soon. in the mean time, hopefully someone can learn a bit more about vector spaces with this thread.

stay tuned

 

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a couple of weeks ago there was a thread here about Stoke s theorems. one of the prerequisites for understanding how to calculate with stoke s theorems is the knowledge of differential forms. so this thread will be the quick and dirty intro to differential forms. differential forms are used in every branch of physics, well, at least by mathematical physicists. once we have finished this thread on differential forms, we ll go back to the stoke s theorem thread, to finish up stoke s theorem in full generality.

i m hoping to keep the prerequisites for this thread at a minimum. you definitely need to know some calculus to follow this thread. you have to know what a derivative is, and you should probably know a little bit of multivariable calculus, although honestly there won t be that much actual calculus used.

also, some familiarity with vector spaces is useful, although everything we need to know about vector spaces is included somewhere. if you are familiar with vectors in ℜ³, and the vector cross product, that will probably suffice. oh and i assume that you know what it means for vectors to be linearly independent, and what a basis of a vector space is. those concepts could certainly be explained here, if anyone wishes.

i do make quite a few references to the concept of a manifold throughout. i don t expect you to know the technical definition of a manifold. we won t need it here, and it is not really taught in any undergraduate math or physics curricula, as far as i know. so let me just describe generally what i mean here when i say manifold.

a manifold is basically a generalization of ℜⁿ. for example, ℜⁿ is itself a manifold, albeit a flat one, but we want to extend our idea of a space to include curved spaces, so lemme just give a few examples: a parabola is a curved 1 dimensional manifold, that extends to infinity. a circle is a 1 dimensional manifold that folds back on itself. a sphere is a 2 dimensional manifold, or actually, any surface you would want to think of is a manifold. a manifold is just a space that is not necessarily flat. that is about all we need to know about them.

i encourage anyone to ask questions about any parts of this thread that are unclear, if you re interested to learn this stuff.

also, there is a lot of text here, and as i reread what i have written, i frequently add, revise, clarify, and correct things, so it might help to occasionally reread the thread, if you notice that i have edited it recently.

 

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Euclidean Vectors

i am going to assume that you are a little familiar with euclidean vectors. a euclidean vector is an arrow between two points. it has direction and magnitude¹. mathematically, we can specify a vector in euclidean space with a pair of points in the space, and let the vector be the arrow directed from one point to the other. or you can assume that the first point is always the origin, and specify the vector with just a single point. by doing this, you are essentially moving the vector from it s basepoint, to the origin. this is possible because euclidean space is both a manifold and a vector space.

this won t be true when we move to noneuclidean manifolds. for example, there is no sensible way to make points on a sphere into a vector space. there is no sensible way to define addition on these points.


<hr>
¹<font size=1>Well, the vectors don t have magnitude or direction until we endow the space with a metric. almost everything we are going to talk about here is independent of metric, and we will not need to specify a metric on this space. when using metric dependent quantities, this is the differential geometry, and when dealing with the more general metric independent quantities, this is differential topology. if you don t know what any of this means, ignore it.</font>

 

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now, hopefully we are all pretty comfortable with what a normal euclidean vector is. it s basically just an arrow between two points. it has a magnitude and a direction. right? euclidean space also comes endowed with a way to calculate the length of a vector. the pythagorean theorem. hopefully we re all familiar with this concept. if anyone wants to hear a little more about euclidean vectors, just holler, and we ll let you know.

 

Abstract Vector Spaces

The notion of vector arithmatic and linear spaces turns out to be very useful in many different areas of mathematics and physics. so let s write down those mathematical properties of vector spaces that make them useful, and forget any notion of vectors as arrows with direction and magnitude. let v and w be vectors, and here i mean it in the abstract sense.

in other words they are just elements of a set which i am going to call vectors. they are not necessarily arrows. let a and b be real numbers². these are the properties that the vectors must satisfy to be called a vector space.

Abelian Group Properties

<ol>
<li>v+w = w+v. i.e. vector addition is commutative.</li>
<li>there is a vector in my set 0 such that v+0=v.

in other words, there is a zero vector</li>
<li>for any vector v, -v is also a vector.
</ol>

Distributivity and Associativity
<ol>
<li>(a+b)v=av+bv</li>
<li>a(v+w)=av+aw</li>
<li>a(bv)=(ab)v</li>
</ol>

this all seems a little abstract, but i assure you, the abstraction will pay off, when we can use all the theorems we know for vectors for all kinds of things that look nothing like our euclidean arrows with magnitude and direction.

any set of objects which satisfy these axioms, together with the set of numbers, is called a vector space over those numbers.



<hr>
²<font size=1>To be abstract and general, i do not need to require that a and b are real numbers. they can be members of any field. in fact, to be completely general, most, but not all, of the useful properties of a vector space also hold if i use a ring, instead of a field. this is called a module, instead of a vector space. if you don t know what any of this means, then ignore it.</font>

 

Dual Space

a linear functional is simply a function that takes vectors as input, and spits out numbers as output. it should also be linear, that is to say, &sigma; is a linear functional over a vector space if and only if
&sigma;(av+bw)=a&sigma;(v)+b&sigma;(w)
where a and b are numbers, and v and w are members of the (abstract, i.e. not necessarily arrows) vector space.

now the reason i talked at length about abstract vectors is that i want you to be comfortable with the fact that something doesn t have to be an arrow to be something i would want to call a vector. polynomials are vectors in the abstract sense, and so are linear functionals. this fact is of paramount importance for our purposes in this thread.

i will ask for a volunteer to show that the set of linear functionals on a given vector space is itself a vector space. it s not to hard, just check the vector space axioms given above.

the set of linear functionals on a vector space V is called the dual space to that vector space, and it is denoted with the symbol V*.

OK, so if you believe me that the dual space of a vector space is itself a vector space, then you should know that the dual space should have a basis. well it does. in fact, there is a special basis for the dual space called the dual basis, that i want to look at now.

suppose we are given a basis for our original vector space V, {e_&mu;}. then this induces a natural choice of basis for the dual space V*, {&sigma;^&nu;}, determined by
&sigma;^&nu;(e_&mu;) = &delta;^&nu;_&mu;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)
in other words, for each basis vector, there exists exactly one linear functional that takes that basis vector to the number 1, and takes every other basis vector to the number 0. these linear functionals form a basis of the dual space, that we will have occasion to use.

the &mu; in {e_&mu;} is just a label that runs from 1 to n, where n is the dimension of the vector space. so that is a set of n independent vectors. likewise, the &nu; in {&sigma;^&nu;} is just an index that runs from 1 to n, where n is the dimension of my dual space, which is the same as the dimension of the vector space. &delta;^&nu;_&mu; is the kronicker delta. it is 1 when &mu; = &nu;, and 0 when &mu; &ne; &nu;. so this just a mathematical symbol that means what i said in words: the i^th basis linear functional has the value 1 when it acts on the i^th basis vector, and has the value 0 when it acts on any other basis vector.

i invite anyone to try, as an exercise, to show that these linear functionals are are unique, and independent, and span the dual space, i.e. that they are a basis as i claim they are. it s not hard, everything follows from the linearity.

some examples: if your vector space is a set of column vectors, then the dual space is the set of row vectors. a row vector operates on a column vector linearly, and yields a number. if your vector space is some quantum mechanical hilbert space, then the dual to the space of kets is the space of bras.

 

Tangent Vectors


nota bene: starting with this post, we will depart from what you learned in your college linear algebra courses, so start paying attention.

OK, so i mentioned that on general manifolds, it makes no sense to think of the points in the space as a vectors. there is simply no way to define addition of points on a sphere in such a way that the abstract vector axioms are satisfied. but what any smooth manifold will have is something called tangent vectors. but we are going to define tangent vectors without any reference to the ambient space.

let me explain what i mean by that last statement in a little more depth. consider the circle, drawn in &real;². we can use our well-known calculus in &real;² to calculate an &real;² vector that is tangent to the circle.

but what if i weren t allowed to draw my circle in &real;²? how would i then write down the tangent vector? this is a subtle point, that some people have trouble with, so if you don t know what i mean here, feel free to ask questions. have you heard people talk about how our 4 dimensional spacetime is curved, but you wondered "what does it curve around?" or "what direction does it curve in?", well that s what i m talking about.

well anyway, let s get on with it. the way we define tangent vectors is through directional derivatives. the formula we learned in regular calculus for a directional derivative is v&sdot;&nabla;&fnof;. we re going to use the same concept to define a tangent vector to a manifold. remember, a manifold, by definition, is equivalent to some &real;ⁿ, so we can always introduce coordinates near a point, and then take derivatives along the lines of thhose coordinates, and never make reference to any ambient space.

at this point, i will stop using classical vector notation. i will write the directional derivative as v^&mu;&part;_&mu;&fnof;, where &part;_&mu; is shorthand for &part;/&part;x^&mu;, and x^&mu; is one of the coordinates, and &mu; is a number that ranges over the number of dimensions of the manifold, from 0 to n-1 usually. so there will be n different coordinates for an n dimensional manifold. and v^&mu; is going to be associated with the &mu;-th component of the vector, to be defined. and even though i didn t write it, i meant for that to be a summation: v&sdot;&nabla;&fnof; = &sum; v^&mu;&part;_&mu;&fnof; = v^&mu;&part;_&mu;&fnof;. i just leave off the &sum; from now on. every time you see an equation with the same letter as a superscript and a subscript, you should sum over that index.

OK, now the meat of the post: we are going to say that a tangent vector is an operator that takes a function, and returns a directional derivative. this is not too hard to understand. given some function on the manifold (doesn t matter what the function is), one way to characterize tangent vectors is to say they specify a direction along the manifold at an instantaneous point. and one way to characterize directions is by directional derivatives. that means, that for every direction, you will get a different value for the directional derivative, given some function. the notion of how much a function changes in any particular direction is independent of the function itself: different functions might increase at different rates in a given direction, but that value is just a linear differential operator on the function. since all the directional information is encoded in that operator, that is what we will want our vectors to be.

we define the vector to be that operator. this is how it operates on a function:
v(&fnof

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= v^&mu;&part;_&mu;&fnof;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)
since this is independent of the function that i want to operator on, let me just write the vector operator:
v = v^&mu;&part;_&mu;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3)
and this is the point of this post. a tangent vector is defined to be/associated with/thought of as a differential operator. v is the vector, and v^&mu; are the coordinate components of the vector, and &part;_&mu; are the coordinate basis vectors of the tangent space. the vector itself is coordinate independent, but the components are not, and the basis vectors are not (that sounds a little redundant, eh? the basis vectors are not independent of the basis vectors. heh. fuck off.)

OK, it should be easy to show that the set of tangent vectors, thusly defined, satisy the axioms of the vector space. i will call this vector space TM_p. that is, the tangent space to the manifold M at the point p is TM_p. for an n dimensional manifold, the tangent space is always an n dimensional vector space.

this should make some sense, because on a curved manifold, you can only consider directions between two infinitesmally close points: the arrow pointing between to finitely seperated points on, say, a circle, is not a tangent vector to the circle, only infinitely close points determine a tangent vector. to determine tangent vectors between two infinitesimally close points, you have to take a limit, and you will end up with a derivative.

nevertheless, a lot of people have a hard time swallowing this equation, including me when i first learned it. why are coordinate derivatives vectors? well, let me just say, think carefully about what s written here, and please, ask questions. it s subtle, and if you can t really convince yourself of why, then just take it as given, so that you can procede with the rest of the thread.

 

1 Forms


i realized that i needed to say a few more things about dual spaces that i had forgotten to mention. so if you re following at home, you might want to go re-read the post on dual spaces, cause we re now going to talk about the dual spaces quite a bit here.

OK, at this point, we are ready to introduce the first kind of differential forms, the 1 forms. a 1-form is simply a member of the dual space to the tangent space at a point.

if M is our manifold, then TM_p is the tangent space at the point p, and T*M_p is the dual space to that tangent space, according to the notation we introduced above for dual spaces.

i will sometimes call a member of the dual space of the tangent space a cotangent vector, and call the dual space itself the cotangent space. thus, a 1-form is simply a cotangent vector.

now, let s recall what a member of the dual space is: it s a linear functional on the vectors. that means that if i operate a member of the dual space on the vector at a point, then i get a number.

but i also recall that i defined the vector space itself as a set of operators! the vectors take functions on the manifold and return the value of the directional derivative at that point. which is also a number! in fact, since the differential operator that defines the vector is a linear operator on functions, i can make a parallel between the functions on the manifold, and the linear functionals in the dual space.

let me explain that a little further: a linear functional on the vector space says "take a vector, spit out a number". a vector says "take a function on the manifold, spit out a number". so for each function on the manifold, i can assign to it a linear functional that spits out the same number when it acts on the vector, as the vector spits out when it acts on the function. for any function &fnof;, let me call this associated linear functional d&fnof;. the d will come to have a familiar meaning, but for right now, it just means find the linear functional who spits out the number required. it s just a symbol that means "the linear functional associated with the function &fnof;".

read that paragraph again, and see if you can follow it. let me write down what i said above, using the symbols we have introduced:

d&fnof;(v) = v(&fnof

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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(4)

on the left hand side, we have a linear functional in the dual space acting on a vector in the tangent space, and on the right hand side, we have that same vector , it remembered that in addition to being a vector in the tangent space, it is also a differential operator, and as a differential operator it is acting on the function associated with my linear functional.

the linear functional takes the vector to the same number that the vector takes the associated function.

be careful of my use of the words function and functional. there isn t any deep difference between the two words, they are just usually used in different contexts. the word functional is usually reserved for mappings that act on vectors or more complicated objects, and functions usually act on numbers.

so d&fnof; is a functional that acts on vectors, and &fnof; is a function, that acts on numbers. (well actually, in our case, it acts on points in our manfold M.)

sorry if i m getting repetitive here, but this is important, and i want to make it clear.

OK, so lets explore some properties of these 1 forms. first of all, let s write down the dual basis, {&sigma;^&nu;}. these are, by definition, linear functionals such that &sigma;^&nu;(e_&mu;) = &delta;^&nu;_&mu; (eq. (1) above). where {e_&mu;} is the basis for the vector space. but remember, for the tangent space, we already chose a basis, the coordinate basis {&part;_&mu;}. also, like we discussed above, our dual space linear functionals on the tangent space can be associated with functions on the manifold. so let s do that for each &sigma;^&nu;: &sigma;^&nu; = d&fnof;^&nu;, where &fnof;^&nu; is some function on the manifold that we will determine.

with these changes, let s rewrite that condition for the dual basis (1):
d&fnof;^&nu;(&part;_&mu;) = &delta;^&nu;_&mu;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(5)

now using (4) above, this becomes:

d&fnof;^&nu;(&part;_&mu;) = &part;_&mu;&fnof;^&nu; = &part;&fnof;^&nu;/&part;x^&mu; = &delta;^&nu;_&mu;

now, can you think of a function whose derivative with respect to x^&mu; is 1, and whose derivative with respect to all other coordinates is 0? it s easy..

think about it....

got it?

it s &fnof;^&mu; = x^&mu;! no sweat!

OK, so then the dual basis of the 1 forms is just {dx^&nu;}. now let s check what the components of a general 1 form are in terms of this basis:

d&fnof; = &alpha;_&nu;dx^&nu;

where &alpha;_&nu; are the components of the 1 form. let s solve for those components by acting this 1 form on a basis vector of the tangent space.

d&fnof;(&part;_&mu;) = &alpha;_&nu;dx^&nu;(&part;_&mu;)
&part;_&mu;&fnof; = &alpha;_&nu;&part;_&mu;x^&nu; = &alpha;_&nu;&delta;^&nu;_&mu; = &alpha;_&mu;

i have used (4) twice in the second equation there.

how about that! the components of the 1 form &alpha;_&nu; are just the partial derivatives of the function!
d&fnof; = &sum;(&part;&fnof;/&part;x^&nu;)dx^&nu; = &part;_&nu;&fnof;dx^&nu;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(6)

now that equation should look familiar perhaps to some of you from your calc classes. it s just the chain rule of multivariable calculus, or at least it looks like it. this explains why we used the symbol "d" to create a 1 form out of a function, because it is done by simply differentiating. at first "d" was just a symbol to associate a linear functional with a function on the manifold. but now we see that it is actually a differential operator on the functions. this "d" operator we will see again, it is called the exterior derivative. it is very important. it is the "d" that appears in the integrand of stoke s theorem.

OK, so that s 1 forms! you folks with me so far? feel free to ask some questions, or if any of you wants to clarify any points that you think i didn t make very well, feel free.

next up, higher order forms!



Edit: added the last term to equation 6, reverting to the more compact notation, for the sake of consistency 5.29.03

corrected TM_p* to T*M_p 5.31.03

 

I ve been using quite a lot of maths symbols in this thread. how many of you that are following the thread are not able to see the symbols correctly? Ron, i know you are having trouble with a lot of these symbols right?

most of these symbols don t work for me on my work computer with the <a href="http://www.konqueror.org/">konqueror</a> for the KDE platform. so if you re using that browser, consider launching something else if you want to read this thread.

if you re using IE for windows, or netscape 4.x on any platform, i am going to highly recommend you switch to <a href="http://www.mozilla.org/projects/phoenix/">phoenix</a>. it is a great browser. you can run it on windows and most *nices. if you re on a mac, this thread works great in omni, safari, and camino. no probs there.

 

Any chance of making this a 'sticky'- I have several questions that I'm sure Lethe can answer - but I don't want the thread to disappear.

Ron.

 

Tensor Products of 1-forms


First, let me tell you what a tensor is. basically, a tensor is either a tangent vector, or a linear functional, i.e. a 1 form. i will use the word as a single word that encompasses both of those notions, which i defined above, as well as certain composites that i will define now.

remember, a 1 form is a functional that takes 1 vector, and spits out a number. if i have 2 1-forms, say &omega; and &sigma;, then the tensor product of these two 1-forms is a new tensor that takes two vectors and spits out a number. it does this by feeding the first vector to the first 1-form, which gives you a single number, and feeding the second vector to the to the second 1-form, and getting another single number, and then returning the product of those two numbers. the notation usually used for this tensor product of 1-forms is &omega;&otimes;&sigma;. so then value of the tensor product, acting on the vectors v and w i just described can be written with my symbols:

&omega;&otimes;&sigma;(v,w) = &omega;(v)&sigma;(w)

on the left hand side of the equation, i show you the tensor product of two 1-forms, acting on an ordered pair of vectors, and on the right hand side, i act the two 1-forms individually on the two vectors, and multiply the 2 numbers that come out.

and that s all there is to the tensor product! pretty simple.

 

Wedge Product


Now, we must define the wedge product of two 1-forms. if you followed my definition of the tensor product, this will be easy:

&sigma;&and;&omega; = &sigma;&otimes;&omega; - &omega;&otimes;&sigma;

easy enough. let s check how that new wedge product acts on our two vectors:

&sigma;&and;&omega;(v,w) = &sigma;(v)&omega;(w) - &omega;(v)&sigma;(w)

one obvious property of this new product is that it is alternating, which means that if you switch the order of the two vectors that you feed to it, you pick up a minus sign from the original product before you switched:

&sigma;&and;&omega;(w,v) = &sigma;(w)&omega;(v) - &omega;(w)&sigma;(v) = &omega;(v)&sigma;(w) - &sigma;(v)&omega;(w) = -(&sigma;(v)&omega;(w) - &omega;(v)&sigma;(w)) = -&sigma;&and;&omega;(v,w)

also, the wedge product is itself antisymmetric, meaning that if you switch the order that you multiply the to 1-forms in, you again pick up a minus sign:

&omega;&and;&sigma; = &omega;&otimes;&sigma; - &sigma;&otimes;&omega; = - (&sigma;&otimes;&omega; - &omega;&otimes;&sigma

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= -&sigma;&and;&omega;

compare this with the tensor product: if you switch the order of the two vectors you input, you get a new number with no general relationship. and if you switch the order of the two 1-forms that you re taking the tensor product of, you get a new tensor that is in no general way related to the original tensor.

anyway, i define a 2-form to be the wedge product of two 1-forms. we can get to a p-form by simply taking the wedge product of p 1-forms, using these definitions.

next up, we will investigate a few more of the algebraic properties of the wedge product.


Edit: changed the first use of term antisymmetric to the term alternating, to remove ambiguity. antisymmetric means change the order of the multiplication, and you get a minus sign. alternating (or symplectic) means change the order of the arguments, and you get a minus sign 5.31.03

 

Old Tensor Notation


at this point, i think i should make some contact with the tensor index notation, and way of doing differential geometry.

in most physics classes, they first introduce vectors on a noneuclidean manifold by looking at how dx^&mu;/dt transforms when you change to a different coordinate system. it looks like this: dx^&mu;&prime;/dt = (&part;x^&mu;&prime;/&part;x^&nu;&prime

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dx^&nu;/dt, by the chain rule. they then declare that any set of n components that transforms the same way is a contravariant vector.

similarly, one then looks at the quantity &part;&fnof;/&part;x^&mu;. it transforms like: &part;&fnof;/&part;x^&mu;&prime; = (&part;x^&nu;/&part;x^&mu;&prime

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&part;&fnof;/&part;x^&nu;

that is to say, it transforms oppositely from a contravariant vector. one then declares any set of components that transform according to this rule, a covariant vector

this is a braindead approach, and i ll tell you why. think back on the first time you learned about euclidean vectors. for us, it was around age 16 or 17 in high school. what was the first thing they told you? " a vector is an arrow. it has a magnitude and a direction." imagine if they had instead told you that a vector is any list of points that transform according to a certain obscure rule when you rotate them? do you think you would have understood that? would you have gained an intuitive physical notion of what a vector is? i don t think so. sure, you could have learned the rules for its manipulations, but you would never have understood those manipulations, until you came to the following realization: requiring that a vector transform thusly under coordinate transformations is equivalent to requiring that its physical meaning is independent of what coordinates you use.

defining quantities in terms of coordinate representations obscures the meaning of the object. for example, when electrodynamics was first being developed, vector calculus as we know it did not exist yet. instead of being able to write things like &nabla;&sdot;E = &rho;/&epsilon;₀, you had to write instead &part;E_x/&part;x + &part;E_y/&part;y + &part;E_z/&part;z = &rho;/&epsilon;₀. i believe that these lengthier coordinate expressions obscured the physical meaning significantly. the introduction of vector calculus allows you to think of vector fluxes, and divergences as sources or sinks, and curls as rotating segments of the field. theorems are easier to prove, and the physics is easier to extract and extend from the equations. i can t tell you how many times i have had people ask me what the difference is between a second rank tensor, and a 3x3 matrix. of course it s not clear what the difference is, when a tensor is just a set of numbers. a tensor is properly defined in terms of it s physical meaning.

this is not to say that i think coordinate representions are bad or useless. they certainly have their place. in fact, you can t really do calculations or measurements in the lab without first choosing a coordinate system. but quantities that are independent of the coordinates should be defined without reference to the coordinates.

let me rant about the tensor index mathematics a little bit longer. another problem that arises from defining your vectors as a set of components is that the dfference between covariant and contravariant vectors becomes very unclear. it is just a slight reordering in the change of coordinates formula, and it is very easy to forget or fuck up.

as we will see, there is a much more natural way to see the difference between covariant and contravariant vectors. it won t have any reference to coordinate bases or transformations, and it will make explicitly clear where the words contravariant and covariant come from. i will make this more precise later, but it will just be a statement like covariant vectors get pushed forward by manifold mappings, and contravariant vectors get pulled back. and that s all. mappings on manifolds act backwards on contravariant vectors.

the important thing that we will see is: everyone gets it wrong! people call tangent vectors are covariant, and yet every physics book will tell you that it is contravariant! what they mean, of course, is that the components of a covariant vector transform just like the corrdinate basis contravariant vectors. but this terminology is decidedly misleading! the components of a vector are themselves not a vector at all! it hardly makes any sense to compare components, which are simply numbers, to basis contravariant vectors, which are linear functionals.

therefore, a lot of books will simply refuse to use the terms covariant and contravariant. modern mathematical physicists are trying to phase out that terminology, and that way of learning physics. and this is the last time you ll hear me mention them in this thread.

in this thread, the components of a tangent vector, when we have occasion to use them, will have superscripts, and the components of a 1-form will have subscripts, if you have learned tensor index notation before, tangent vectors are going to replace contravariant vectors, even thought tangent vectors are covariant, and 1-forms are going to replace covariant vectors, even though they are contravariant.

 

JR,

Can you make this a 'sticky' and perhaps take out the non-essential chat?


With a bit of luck, Lethe will get a book out of this!

cheers,

ron.

 

Lethe, please don't get sidetracked by other threads that appear every fukin day or could be answered by a cursory glance into any reasonable intro text (not trying to be elitist - just trying to learn some physics).

cheers,

Ron.

Just my attempt to keep this up thread up there (JR - I'm not a mod - can I make this a sticky? It's about the only non-crackpot physics that has lasted for any time).

 

lethe,

had saved this page already. whenever u modify pl mark it by colored postscript number to know it easily. i don't find any reason why this thread can't be made sticky. atleast another sticky-thread that contains link to this thread and similar informative threads with a brief.

 

that s a good idea. i don t think i will go back and do that now, because i can t really remember all the changes i ve made, but in the future, if i make some major changes to any of the posts, i will mark that in color. i do go back and make minor corrections, fix grammar, typos and such, and i don t think i will make color changes for things like that.

for example, just today, i realized something: the html code for the script &fnof; is &amp;fnof;. if i enclose that in parentheses, the board turns it into a smiley, instead of an &fnof;. which is annoying. so i was leaving off the final semicolon, which fixed the problem in my browser, but i just realized that the symbol was broken in IE this way. so i went back and put those semicolons back in, and checked off the "disable smileys" option. this is a much better solution, and i did that to several posts just now. now i can t use smileys, but i don t really use those anyway, so i don t care.

yes: this forum needs a FAQ. that would be sticky. it could tell newcomers what is and is not appropriate to this forum. then it could link to a couple of threads. like, one that talks about relativitiy, so that we don t have a new relativity thread every single week, we could keep that all in one place. one for Xeno s paradox might be nice. it could tell people the difference between rest mass and relativistic mass. it could link to a thread about quantum indeterminacy. it could list what maths is required for physics, that s another one that gets asked a lot.

and then, there could be a sequence of survey threads for all the maths subjects required. this could be the differntial forms thread. we would need a calculus thread, an algebra thread, a differential geometry thread (maybe this thread can grow into a differential geometry thread eventually), maybe a topology thread too. only problem is, most of those threads are not written, and they would be a lot of work on someone s part. we would have to basically write down an entire curriculum for an undergraduate maths degree.

still, i think it is a good idea, it could make this forum a much more professional looking place. the more maths we teach, the more fruitful our discussions of physics can be.