Quick question, the context is NMR, for the case of spin 1/2 particles. The I-operator is the angular momentum operator. They introduce a unitary transform which represents a rotation around the z axis, so it has indeed the form of an exponential containing the angular momentum operator and some angle. The fancy 1 is the unity operator/matrix.
[IMG]http://quicklatex.com/cache3/ql_d715a7ae2b76f10f59b675163ef5a952_l3.png[/IMG]
Does anyone have a clue which identity was used here? I tried the Baker–Campbell–Hausdorff formula but that wouldn't explain the cosine & sine. It has hints of De Moivre (although I wouldn't know how to apply it to a matrix exponential) and some half angle formula.
So in short, given that exponential definition of U, how do you change it into the first line.
It's mentioned on [URL="http://www.physics.ox.ac.uk/users/iontrap/ams/teaching/rel_C_spinors.pdf"]page 9 here[/URL] too.
Calculus for only 7 more weeks, i better pass it this time around
[QUOTE=Number-41;47872218]Quick question, the context is NMR, for the case of spin 1/2 particles. The I-operator is the angular momentum operator. They introduce a unitary transform which represents a rotation around the z axis, so it has indeed the form of an exponential containing the angular momentum operator and some angle. The fancy 1 is the unity operator/matrix.
[IMG]http://quicklatex.com/cache3/ql_d715a7ae2b76f10f59b675163ef5a952_l3.png[/IMG]
Does anyone have a clue which identity was used here? I tried the Baker–Campbell–Hausdorff formula but that wouldn't explain the cosine & sine. It has hints of De Moivre (although I wouldn't know how to apply it to a matrix exponential) and some half angle formula.
So in short, given that exponential definition of U, how do you change it into the first line.
It's mentioned on [URL="http://www.physics.ox.ac.uk/users/iontrap/ams/teaching/rel_C_spinors.pdf"]page 9 here[/URL] too.[/QUOTE]
Looks like some sort of Eulers formula too.
Maybe the Iz operator works only on imaginary numbers, not affecting real part, so that is why it's 1 there?
I think I can do it explicitly by using Pauli matrices.
Ok I found it [URL="http://grothserver.princeton.edu/~groth/phy505f12/sol08.pdf"]here[/URL], I forgot a factor. This is a bit of a sloppy course though, they use identities that are only defined a couple of pages further on.
Can epsilon-delta limit be written as:
where
lim ƒ(x) = L
x->c
∀ ε > 0, ∃ δ > 0, such that
L - ε < ƒ(x) < L + ε -> c - δ < x < c + δ
Can anyone here deal with a Physics w/ Calculus course?
[QUOTE=proboardslol;47884486]Can epsilon-delta limit be written as:
where
lim ƒ(x) = L
x->c
∀ ε > 0, ∃ δ > 0, such that
L - ε < ƒ(x) < L + ε -> c - δ < x < c + δ[/QUOTE]
Pretty much, though I think you want the statement that for all x in delta range of c, you want f(x) to be in epsilon range of L, and to me your statement reads the opposite way around if you take the -> to mean implies, may be wrong though.
[QUOTE=Rct33;47884606]Pretty much, though I think you want the statement that for all x in delta range of c, you want f(x) to be in epsilon range of L, and to me your statement reads the opposite way around if you take the -> to mean implies, may be wrong though.[/QUOTE]
so...
[code]c - δ < x < c + δ -> L - ε < ƒ(x) < L + ε[/code]
?
[QUOTE=The Robster;47884554]Can anyone here deal with a Physics w/ Calculus course?[/QUOTE]
Yes.
Anyone here studied any differential algebra? I'm thinking about doing my masters thesis on basic differential algebra and galois theory. I'll probably be going through "An Introduction to Differential Algebra" by Kaplansky to get started with the field. Seems like a pretty good book to me. Any opinions?
I'm going into my state university soon for Computer Engineering. I need to determine what math I should be taking. Math 141/calculus supposedly, which is the course after pre-calculus here, but the problem is that I'm 21. I've waited 3 years before going to university since high school, and I fear that I may have forgotten everything (not that my calculus education was very good, anyway). How should I determine what math I'm good for, and what I still remember adequately enough?
[QUOTE=Number-41;47872218]Quick question, the context is NMR, for the case of spin 1/2 particles. The I-operator is the angular momentum operator. They introduce a unitary transform which represents a rotation around the z axis, so it has indeed the form of an exponential containing the angular momentum operator and some angle. The fancy 1 is the unity operator/matrix.
[IMG]http://quicklatex.com/cache3/ql_d715a7ae2b76f10f59b675163ef5a952_l3.png[/IMG]
Does anyone have a clue which identity was used here? I tried the Baker–Campbell–Hausdorff formula but that wouldn't explain the cosine & sine. It has hints of De Moivre (although I wouldn't know how to apply it to a matrix exponential) and some half angle formula.
So in short, given that exponential definition of U, how do you change it into the first line.
It's mentioned on [URL="http://www.physics.ox.ac.uk/users/iontrap/ams/teaching/rel_C_spinors.pdf"]page 9 here[/URL] too.[/QUOTE]
Matrix exponentiation isn't defined unless you use a series expansion.
If you take the Taylor series of the transformation operator about t = 0 you'll find that every odd power in your expansion retains an i, and even powers in your expansion rotate through positive and negative values (due to i^2 = -1 and i^4 = 1). If you then separate out the odd and even terms of your expansion and pull 'i' out to the front of the odd terms (well, 2i in this case) you'll realise that you have the Taylor expansion of the sin function, and the even terms turn out to be the Taylor series expansion of the cosine function.
[QUOTE=bitches;47888708]I'm going into my state university soon for Computer Engineering. I need to determine what math I should be taking. Math 141/calculus supposedly, which is the course after pre-calculus here, but the problem is that I'm 21. I've waited 3 years before going to university since high school, and I fear that I may have forgotten everything (not that my calculus education was very good, anyway). How should I determine what math I'm good for, and what I still remember adequately enough?[/QUOTE]
At the beginning of the term, there might be a math placement test which answers this question to the satisfaction of your school.
Before then, you can do these:
[url]http://www.csuohio.edu/sites/default/files/SAMPLE_EXAM_PLACEMENT_Without_solutions.pdf[/url]
[url]http://www-math.umd.edu/cgi-bin/placement/index.cgi[/url]
In addition, take a look at various online precalculus resources (e.g. For Dummies, khanacademy) and the text book for Math 141 (provided the bookstore will let you return it at the price you paid).
[QUOTE=Gas/spg;47890824]At the beginning of the term, there might be a math placement test which answers this question to the satisfaction of your school.
Before then, you can do these:
[url]http://www.csuohio.edu/sites/default/files/SAMPLE_EXAM_PLACEMENT_Without_solutions.pdf[/url]
[url]http://www-math.umd.edu/cgi-bin/placement/index.cgi[/url]
In addition, take a look at various online precalculus resources (e.g. For Dummies, khanacademy) and the text book for Math 141 (provided the bookstore will let you return it at the price you paid).[/QUOTE]
Dover books are fucking fantastic value for money and if you're prepared to spend some time going through one and do the questions/worked examples present throughout then it should help prepare you very well; I'd advise those to him over what will probably wind up being a few hundred dollar text book.
I've heard Schaum's Outlines are also fantastic.
[QUOTE=sltungle;47890956]Dover books are fucking fantastic value for money and if you're prepared to spend some time going through one and do the questions/worked examples present throughout then it should help prepare you very well; I'd advise those to him over what will probably wind up being a few hundred dollar text book.
I've heard Schaum's Outlines are also fantastic.[/QUOTE]
Dover books are great. Gelfand and Fomin is probably the best mathematical introduction to the calculus of variations that there is and it's like $20.
[I]A Student's guide to <topic>[/I] published by CUP are also excellent introductory texts for a lot of subjects. I found the Lagrangians and Hamiltonians as well as the Maxwell's Equations books quite helpful in my first treatment of those topics.
I'm working on signing up for courses next semester. I'm a transfer student to NYU and I feel like I'm scraping the bottom of the barrel in terms of open courses.
What did you guys think of your undergrad geometry courses? I'm a little offput because I'm hearing from some people that it's easy.
Also, how was Abstract Algebra? I'm actually terrified of that one, because I'm hearing it's going to rip my asshole asunder.
[QUOTE=Xeloras;47917562]I'm working on signing up for courses next semester. I'm a transfer student to NYU and I feel like I'm scraping the bottom of the barrel in terms of open courses.
What did you guys think of your undergrad geometry courses? I'm a little offput because I'm hearing from some people that it's easy.
Also, how was Abstract Algebra? I'm actually terrified of that one, because I'm hearing it's going to rip my asshole asunder.[/QUOTE]
Are we talking basic Euclidean geometry such as Euclid's postulates and the geometry of triangles (maybe a sprinkling of spherical and basic hyperbolic in there) or more advanced stuff like differential geometry?
The former isn't too bad, such a course is usually quite easy to follow. The latter is much more indepth and proof based. You will have to have a good understanding of calculus and linear algebra.
Abstract Algebra looks tough when you first get into it but it really isn't too bad once you understand what is going on. They key to passing a basic proof based course (such as abstract algebra or introductory real analysis) is to make sure you [i]understand[/i] the definitions. Many people approach these kind of courses by just memorising definitions but that won't help much. You really need to get knee-deep in the material; making sure you are able to follow each and every logical step in the proofs without issues.
95% of teachers will not ask you to reproduce crazy long (or hard) proofs. The aim of these courses is not to test your memory, rather your ability to reason with the material you have learnt. That being said, it is very good practice to go through the proofs in your notes one by one, making sure you can write each one out without having to refer back. This may seem like memorisation at first but it actually helps in cementing the logical steps and furthers your understanding of how the proofs work. By doing this, you will gain intuition in how a proof is constructed.
I think I might have digressed a little but the point is that you shouldn't be afraid of these maths courses. If you dedicate time to understanding the material, you will go far.
[QUOTE=Xeloras;47917562]I'm working on signing up for courses next semester. I'm a transfer student to NYU and I feel like I'm scraping the bottom of the barrel in terms of open courses.
What did you guys think of your undergrad geometry courses? I'm a little offput because I'm hearing from some people that it's easy.
Also, how was Abstract Algebra? I'm actually terrified of that one, because I'm hearing it's going to rip my asshole asunder.[/QUOTE]
I didn't even have an undergrad geometry course. What does it cover?
I like abstract algebra and I find the proofs to be usually fairly simple and intuitive. Don't let people who don't like the subject scare you into thinking it's hell for everyone.
I guess we're not learning l'hopitals rule in this calc class. Strange
[editline]10th June 2015[/editline]
This professor is awesome though so hopefully ill do ok
[QUOTE=cody8295;47925688]I guess we're not learning l'hopitals rule in this calc class. Strange
[editline]10th June 2015[/editline]
This professor is awesome though so hopefully ill do ok[/QUOTE]
I never learnt L'Hopital's rule throughout my undergraduate degree. I don't even think we were allowed to use it in any of my courses. Bit weird if you ask me.
Trying to choose my project topic for my 4th year. Was given a list of 38 possible topics and I've crossed off all but 5:
- Hopf Algebras
- Gelfand Kirillov Dimension
- Fusion in Finite Groups
- Amenable Groups and Semigroups
- Quiver Representations
This is such a difficult decision to make. Whichever I pick I have to spend an entire year writing a project on :(
JohhnyMo1, how come Quaternions are not used for Quantum stuff.. aren't atoms and particles 3D with rotation? I mean it's bit weird Complex numbers are used since they are 2D
What website do you guys point to to learn higher order mathematics online, like differential equations and vector calculus?
Khan Academy, Google, Quora, Math Overflow, Math.StackExchange, Facepunch
[editline]11th June 2015[/editline]
By the way, anyone knows how to make upper triangle matrix with help of gauss?
A = original matrix
P, P^-1 = change of basis matrices
T = triangle matrix
To get something like that?
A = P T P^-1
See [url=http://math.stackexchange.com/questions/281833/matrix-similarity-upper-triangular-matrix]this[/url] answer on stackexchange. It links to the wiki page on the [url]http://en.wikipedia.org/wiki/Schur_form[/url], which also has P unitary.
I keep seeing stuff about convex sets. What is so useful about them?
Can anyone give a brief explanation as to what the members of an infinite Cartesian product are from an axiomatic set theory point of view? I'm used to seeing infinite products indexed over the natural numbers (from basic topology) where elements are just sequences. I'm having a hard time visualising the infinite product over a general indexing set (where members are apparently functions whose domain is the index set).
[QUOTE=JohanGS;47941240]I keep seeing stuff about convex sets. What is so useful about them?[/QUOTE]
You have V_k points. Then you have coefficients C_k. k is index.
V_k € R^3 (or R^2 whatever), C_k € R.
Now you have equation:
C_1 + C_2 + ... + C_k = 1.
That is, sum of all coefficients must be 1 (one).
Then with this equation:
C_1*V_1 + C_2*V_2 * ... * C_k*V_k = point that always lies inside convex set defined by those points.
For example, with three points you form set which represents triangle. With coefficients you can choose any point inside triangle.
It also means nice property, you can choose ANY numbers for C_1, C_2..C_k coefficients, then NORMALIZE them, so sum of them is of 1.. boom, you have point inside convex set.
The properties of convex sets are very useful in optimization problems. If you're trying to find a minimum of a convex function defined on a convex set, then simply finding a local minimum is sufficient, because in that case, any local minima will also be global minima. So basically if you know that both the function and the set over which you're trying to minimize it are convex, your job becomes a lot easier.
[url]https://en.wikipedia.org/wiki/Convex_optimization[/url]
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