I'm currently taking a class called Foundations of Geometry. We started with the stereographic projection and carried onward through fractional linear transformations, and now we are working with the Poincaré Disk Model. We've been finding things like non-Euclidean lines, circles with their non-Euclidean centers and non-Euclidean distances, delving into hyperbolic trigonometry. We don't use a textbook though, the professor just wrote up his own notes, and while good, they're restricted to just our 10 quarter class, and we're just studying one of the non-Euclidean geometries. So I was wondering if anyone had suggestions on some books that may cover more of the hyperbolic geometry or perhaps some elliptic, or any others in general. Thanks! =D
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I'm researching non-Euclidean geometry.  Now I am looking for a good source for it.  Please suggest me something.Thanks.
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I am hoping that someone can help me see understand a proof that the sum of angles of a hyperbolic triangle $\triangle ABC$ is strictly less than $\pi$.  I want to stick to the upper-half-plane model $\mathbb H:=\{z\in \mathbb C : Im (z)&gt;0\}$ of hyperbolic geometry where lines are either vertical lines or semicircles centered on (but excluding) the $x$-axis.  To simply things, by using an LFT we can take one vertex $A$ to be the point $i$ and another $B$ to be $(0,k)\in \mathbb H$ for some $k&gt;0$.    Now, call the third vertex $C$. We have the other two sides $AC$ and $BC$ as arcs of the circles $C_1 =(x-a)^2+y^2=r_1^2$ and $C_2=(x-b)^2+y^2=r_2^2$, respectively, so that $x^2+1=r_1^2$ and $x^2+k^2=r_2^2$.  In my class notes, there is a brief mention of how this simplification can help us with the proof. However, I do not see at all how this helps us get the result that $\triangle ABC$ sums to less than $\pi$. Therefore, any help is very appreciated.  P.S., My knowledge is elementary, so I am hoping for a proof using basic tools.
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I was wondering if you could tell me about some self-learning resources for linear model theories. My professor has been using "A First Course in Linear Model Theory, Ravishanker and Dey. Publisher: CRC." I found this kinda difficult. My major is mathematics. Thank you so much for your time!
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Since summer comes with a lot of spare time, I've decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really caught my eye, but I have no prior training in it.   I've decided to conduct my studying effort in a library; I prefer real books to virtual ones, but as I'm not allowed to browse through the books on my own, I have to know beforehand what I'm looking for and this is where I'm kind of lost. I'm not really sure where to start.  Basically I wanna know about the following:    What are the prerequisites? (I'm currently trained in Linear Algebra, Calculus, Complex Analysis - all on an undergraduate level )  Can you recommend some reading materials?     Thank you.
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I am taking a course in Mathematical Methods for physicsPhysics (Junior Level).   We are working from   > Mathematical Methods for physicists, George B.Arfken .    I just need online resources covering ( Matrices, Determinants, Vector Analysis, Tensors and differential forms and Vector spaces...etc) to study and understand from.
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What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory?  Some subjects off the top of my head that probably need covering:   Differential geometry, Manifolds, etc. Lie groups, Lie algebras and their representation theory. Algebraic topology.
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I know that the geodesics for Euclidean Space are straight lines and likewise in the absence of forces like gravity, the geodesics are straight lines. But what if you took some curved lines and tried to work backwards to determine the geometry of the space consisting of the curved geodesics. How would one go about determining the shape of this space? Would this even be a possible or useful approach?
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What would be your recommendation for the text which could be useful for someone starting with lattice theory? Suggestion for both books and online materials/lecture notes are welcome.    This was asked as a part of  and I did not like that it was combination of two different questions. (One of them on geometric modular lattices and the other one about text recommendation.)  I don't expect the user to post a follow-up question, since he was not seen at MSE for some time. I think that this could be a useful for some users here, so I've decided to post it myself.
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Lifted from :  I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc.  One suggestion per answer please. Also, please include an explanation of why you like the book, or what makes it unique or useful.
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Show that two non intersecting circles can always be mapped by a suitable Möbius transformation to two concentric circles.  I wanted to map the center of the first circle to the center of the second circle then using scaling the radius would give me the concentric circles. But having problem to execute.  Suggestions please,
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Is it possible to get into algebraic geometry by just knowing calculus and linear algebra or is this too far of a stretch?  If not could anyone give me a list of book/lecture notes recommendations in chronological order to dive into algebraic geometry with my state of knowledge?  Thanks
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I need to find a good book on trigonometry, I was using trigonometry demystified but I got sad when I read this line:     Now that you know how the circular functions are defined, you might wonder how the values are calculated. The answer: with an electronic calculator!   I know a book which seems to be really good: Loney's Plane Trigonometry, I'm just not sure if the book is up to date.
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Is there a generalized way( a particular conic section of any shape,for instance an ellipse without determining its major/minor axis) to find the centre,vertices,foci,focal radii,letus rectum,eccentricity,etc when exists of any parabola,ellipse,hyperbola?(dealing separately)  $ax^2+by^2+cx+dy+exy+c=0$ ($a,b,c$ not all zero)
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Just a quick question to verify whether I'm right.   Claim: The fundamental group of the complement of $n$ lines through the origin in $\mathbb{R}^3$ is $F_n$, the free group on $n$ generators.  Proof: remove a line from $\mathbb{R}^3$. We may deformation retract the remaining space onto a cylinder radius $\epsilon$ about the line, and thence to a circle $S^1$. There is no trouble repeating this process with a second distinct line, except that then we will be a wedge union $S^1 \vee S^1$. Continue inductively, and recall that the wedge union of $n$ circles has the stated fundamental group.  I'm only just starting to really get my head around this stuff, so any feedback would be really useful!  Thanks!
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Is there any good book on Electromagnetism from a more mathematical point of view? By this I mean a book which makes use of differential forms and maybe De Rham cohomology. I was also searching for something that would include E&amp;M in the big picture of Gauge theories.
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I'm a PhD student in math and am really excited about celestial mechanics.  I was wondering if anyone could give me a roadmap for learning this subject. The amount of information about it on the internet is overwhelming, and I honestly don't know where I should start.  Any help would be greatly appreciated.
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I'm looking for a book to learn Algebra. The programme is the following. The units marked with a $\star$ are the ones I'm most interested in (in the sense I know nothing about) and those with a $\circ$ are those which I'm mildly comfortable with. The ones that aren't marked shouldn't be of importance. Any important topic inside a unite will be boldfaced.  U1: Vector Algebra. Points in the $n$-dimensional space. Vectors. Scalar product. Norm. Lines and planes. Vectorial product.  $\circ$ U2: Vector Spaces. Definition. Subspaces. Linear independence. Linear combination. Generating systems. Basis. Dimesion. Sum and intersection of subspaces. Direct sum. Spaces with inner products.  $\circ$  U3: Matrices and determinants. Matrix Spaces. Sum and product of matrices. Linear ecuations. Gauss-Jordan elimination. Range. Roché Frobenius Theorem. Determinants. Properties. Determinant of a product. Determinants and inverses.  $\star$ U4: Linear transformations. Definition. Nucleus and image. Monomorphisms, epimorphisms and isomorphisms. Composition of linear transformations. Inverse linear tranforms.  U5: Complex numbers and polynomials. Complex numbers. Operations. Binomial and trigonometric form. De Möivre's Theorem.  Solving equations. Polynomials. Degree. Operations. Roots. Remainder theorem. Factorial decomposition. FTA. Lagrange interpolation.  $\star$ U6: Linear transformations and matrices. Matrix of a linear transformation. Matrix of the composition. Matrix of the inverse. Base changes.   $\star$ U7: Eigen values and eigen vectors Eigen values and eigen vectors. Characteristc polynomial. Aplications. Invariant subspaces. Diagonalization.   To let you know, I own a copy of Apostol's Calculus $\mathrm I $ which has some of those topics, precisely:   Linear Spaces Linear Transformations and Matrices.   I also have a copy of Apostol's second book of Calc $\mathrm II$which continues with   Determinants Eigenvalues and eigenvectors Eigenvalues of operators in Euclidean spaces.   I was reccommended Linear Algebra by Armando Rojo and have Linear Algebra by , which seems quite a good text.   What do you reccomend?