Transformation Definition and 1000 Threads

  1. C

    Help needed for transformation of stresses in beer and johnston book

    Transformation of stresses in beer and johnston mechanics of materials. While reading the section on trsnformation of stresses they have solved by using the force components in x' and y' directions. I have attached a screenshot of the relevant page and the figure. I have few doubt as to how the...
  2. X

    Finding the Standard Matrix A of a Linear Transformation T

    Homework Statement Let T be a linear transformation from R3 to R3. Suppose T transforms (1,1,0) ,(1,0,1) and (0,1,1) to (1,1,1) (0,1,3) and (3,4,0) respectively. Find the standard matrix of T and determine whether T is one to one and if T is onto Homework Equations The Attempt...
  3. S

    Vector Transformation in Cartesian and Polar Coordinates

    Greetings, My question is from the book "Tensor Analysis" by Barry Spain. I am asked to show that how a vector transforms from rectangular Cartesian coordinates to polar coordinates. I have attached the question in jpeg format. I have came up with a solution but the angular component in my...
  4. I

    Complex conjugate as a Mobius transformation

    Hi guys, I am having a very stupid problem. I can't figure out what Mobius transformation represents T(z)=z*, where z* is the complex conjugate of z. In my book we are learning about Mobius transformations and how they represent the group of automorphisms of the extended complex plane (Ʃ). [...
  5. A

    Physical Interpretation of point transformation invariance of the Lagrangian

    Homework Statement The problem asked us to show that the Euler-Lagrange's equations are invariant under a point transformation q_{i}=q_{i}(s_{1},...,s_{n},t), i=1...n. Give a physical interpretation. Homework Equations \frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})=\frac{\partial...
  6. F

    Invariance of a Lagrangian under Transformation

    Homework Statement Show that the Lagrangian \mathcal{L}=\frac{m}{2}\vec{\dot{r}}^2 \, \frac{1}{(1+g \vec{r}^2)^2} is invariant under the Transformation \vec{r} \rightarrow \tilde{r}=\vec{r}+\vec{a}(1-g\vec{r}^2)+2g\vec{r}(\vec{r} \cdot \vec{a}) where b is a constant and \vec{a} are...
  7. F

    Understanding Dirac Delta Function: Time Derivative & Hankel Transformation

    Hi All, I have a problem in understanding the concept of dirac delta function. Let say I have a function, q(r,z,t) and its defined as q(r,z,t)= δ(t)Q(r,z), where δ(t) is dirac delta function and Q(r,z) is just the spatial distribution. My question are: 1. How can I find the time derivative...
  8. P

    Finding a linear transformation.

    Hi, Homework Statement How may I find (or prove that there isn't) a linear transformation which satisfies T: R3->R1[x], ker T = Sp{(1,0,1), (2,-1,1)}? Homework Equations The Attempt at a Solution I am not sure how to approach this. I understand that kerT is the group of all...
  9. I

    Finding T(0,-5,0) from Given Linear Transformation Values

    Homework Statement You are given that T is a linear transformation from R^3 to P2, that T((1,1,-1)) =X, and that T((1,0,-1))=X^2+7X-1. Find T(0,-5,0) or explain why it cannot be determined form the given information. Homework Equations None The Attempt at a Solution There is only X...
  10. J

    Confused about Bogoliubov transformation

    So I was told by my teacher today that I am doing the bogoliubov transformation wrong because I'm supposed to have 4 different vectors, i.e Instead of Ak and Bk A1k A2k and B1k B2k I'm wondering how this makes sense. I've seen papers with 2 different vectors and their complex...
  11. P

    MHB Linear fractional transformation fixing a line

    I'm trying to find the set $\mathscr{F}$ of all linear fractional transformations (l.f.t.) of the unit disc D in itself which map 1 in 1, -1 in -1 and i in -i. By l.f.t. i mean a function$$f(z)=\frac{az+b}{cz+d}$$with $a,b,c,d\in\mathbb C$, $ad-bc\neq0$.I know that this kind of maps sends lines...
  12. P

    MHB What is the image of \(ax+by+c=0\) under \(f\)?

    Consider the affine transformation \(f(P)=\begin{bmatrix}1 & 2 \\3 & 4\end{bmatrix}P+\begin{bmatrix}5\\6\end{bmatrix}\). Find the image of \(ax+by+c=0\) under \(f\). My answer is \(\left(a-\frac{b}{2}\right)y+\left(\frac{3b}{2} -2a\right)x+4a-\frac{9b}{2}+c=0\).
  13. 7

    Lorentz transformation of y cpmponent for 4-momentum

    I have 2 coordinate systems which move along ##x,x'## axis. I have derived a Lorentz transformation for an ##x## component of momentum, which is one part of an 4-momentum vector ##p_\mu##. This is my derivation: \scriptsize \begin{split} p_x &= mv_x \gamma(v_x)\\ p_x &= \frac{m...
  14. 1

    Linear transformation across a line

    Using linear transformation reflection to find rotation Homework Statement Let T1 be the reflection about the line −4x−1y=0 and T2 be the reflection about the line 4x−5y=0 in the euclidean plane. The standard matrix of T1 \circ T2 is what? Thus T1 \circ T2 is a counterclockwise rotation...
  15. M

    Partial derivatives after a transformation

    Suppose I have a transformation (x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2)) and I wish to find \partial x'_1\over \partial x'_2 how do I do it? If it is difficult to find a general expression for this, what if we suppose f,g are simply linear combinations of x_1,x_2 so something like ax_1+bx_2 where...
  16. mnb96

    How the inner product changes under non-linear transformation

    Hi, if we suppose x and y are two elements of some vector space V (say ℝn), and if we consider a linear function f:V→V', we know that the inner product of the transformed vectors is given by: \left\langle f\mathbf{x} , f\mathbf{y} \right\rangle = \left\langle \mathbf{x} ...
  17. B

    Volume of a sphere under a linear transformation R3->R4.

    Homework Statement So there's a linear transformation T: ℝ3 → ℝ4, standard matrix A that satisfies det(A e1) = 5, det (A e2) = 4, det (A e3) = 5 and det (A e4) = 5 If S is the unit sphere, find the 3-dimensional volume of T(S). Homework Equations Volume of sphere = 4/3 * pi * r^3...
  18. J

    Standard Matrix for an orthogonal projection transformation

    Let T:R^2 -> R^2 be the linear transformation that projects an R^2 vector (x,y) orthogonally onto (-2,4). Find the standard matrix for T. I understand how to find a standard transformation matrix, I just don't really know what it's asking for. Is the transformation just (x-2, y+4)? Any...
  19. S

    Derivative Transformation with Law of Cosines

    Hi there, The Law of Cosines can be stated as a^2 = b^2 + c^2 - 2bccos(A) where a,b, and c are the sides of a triangle, and A is the angle opposite the side a. I have a function, f(b,c,A), with an associated set of partial derivatives (\frac{∂f}{∂c})_{b,A} etc. What I want to do is to...
  20. G

    Regular Transformation Homework: Definition & Interpretation

    Homework Statement Hi! i want to ask somebody who are studying quantum mechanics about the definition of regular transformation. I guess there might be people who are not familiar with the notion. So, i'd like to let you know which book I'm referring to; "principles of quantum mechanics" ...
  21. O

    Transformation from cartesian to cylindrical coordinates

    Homework Statement I'm trying to get to grips with Godel's 1949 Paper on Closed Time-like Curves (CTCs). Currently I'm trying to confirm his transformation to cylindrical coordinates using maple but seem to keep getting the wrong answer. Homework Equations The line element in cartesian...
  22. X

    Transformation matrix of linear n-dimensional state-space system

    Hi all, I have a linear algebra question relating actually to control systems (applied differential equations) for the linear system {\dot{\vec{{x}}} = {\bf{A}}{\vec{{x}}} + {\bf{B}}}{\vec{{u}}}\\ \\ A \in \mathbb{R}^{ nxn }\\ B \in \mathbb{R}^{ nx1 }\\ In class, we formed a...
  23. M

    Finding an image under a given transformation

    My question is in the paint document. And I think I know the answer to my question. I asked why can't I let v = 1 then my first first region transformation would the line y = b between -a≤x≤a. The reason I think I can't do this is because the end point v = 1 is a point and not a line...
  24. T

    Linear transformation, subspace and kernel

    Hi We have a linear transformation g : ℝ^2x2 → ℝ g has U as kernel, U: the 2x2 symmetric matrices (ab) (bc) A basis for U is (10)(01)(00) (01)(10)(01)I thought this would be easy but I've been sitting with the problem for a while and I have no clue on how to solve it...
  25. B

    Time reversal transformation of electromagnetic four-potential

    Consider the time-reversal Lorentz transformation given by the 4x4 matrix: \Lambda_T = \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 &1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &1 \end{pmatrix}. In my relativistic quantum mechanics lecture, we discussed how the electromagnetic 4-potential...
  26. H

    Limits? changing and transformation

    okay, so I'm at the most elementary stage of learning limits and there are things which leave me baffled at times, namely two. 1. lim (x -> a) f(x) = lim (x+k -> a+k) f(x) how? the physical reason behind this? 2. the theorem to evaluate limits of the form --- 1^infinity if f(x)=g(x)=0...
  27. N

    Differentiation Map of a Complex Transformation

    Homework Statement Find the eigenvectors and eigenvalues of the differentiation map C1(R) -> C1(R) from the vector space of differentiable functions to itself. Homework Equations The Attempt at a Solution Hi, I'm not entirely sure how to go about this, because would the...
  28. A

    Simplest derivation of Lorentz Transformation

    I'm just getting started on relativity. I watched this a couple of day ago - But I didn't like the way Lorentz Transformation was derived (the assumption about the nature of the final transformations, to be more specific). I tried reading Einstein's original paper for a better derivation but...
  29. N

    Inverse Laplace Transformation of arctan (s/2)

    The Title pretty much says it all. I'm trying to learn how to solve the Inverse Laplace Transformation of Arctan(s/2). An equation of this sort was not explicitly covered in class and I'm having difficulty figuring where to start to solve it. If anyone could give me a general idea that would...
  30. M

    Lorentz Transformation of y-velocity

    Homework Statement A Particle moves with uniform speed V'y = Δy'/Δt' along the y'-axis of the rocket frame. Transform Δy' and Δt' to laboratory displacements Δx, Δy, and Δt using the Lorentz transformation equations. Show that the x-component and the y-component of the velocity of this...
  31. T

    Archived Sign Discrepancy in Plane Stress Transformation Solutions?

    Homework Statement Here is the problem with my attempt at the solution: The magnitude of my answers are correct, HOWEVER I am getting the wrong signs. For the force balance in the x direction I get a negative P but for the force balance in the y direction I get a positive P. Does anyone...
  32. S

    Linear Transformation and Determinant

    Homework Statement Define L: R(mxm) to R(nxn). If L(A)=L(B), prove or disprove that det(A)=det(B). Homework Equations The Attempt at a Solution I think I can prove that this is true. L(A)=L(B) means that L(A)-L(B)=L(A-B)=0. Now let C be the matrix representation of L. We...
  33. A

    Building a rotational matrix transformation

    I am trying to build a rotational transformation matrix both for counterclockwise and clockwise angles. The first matrix in the picture is for counterclockwise angles and the second one for clockwise angles. The first matrix I built corresponds to the one given in my linear algebra book so it...
  34. S

    MHB Linear Transformation (Fredholm Alternative Theorem)

    Let T:V->V be a linear operator on an n-dimensional vector space. Prove that exactly one of the following statements holds: (i) the equation T(x)=b has a solution for all vectors b in V. (ii) Nullity of T>0
  35. S

    What do the eigennumbers of the Lorentz transformation represent?

    I am currently looking a bit into special relativity. Consider the matrix \Lambda=\left( \begin{array}{cc} \gamma & -\gamma \beta c \\ -\gamma \beta c & \gamma \end{array} \right) where \beta=\frac{v}{c},\quad \gamma=\frac{1}{\sqrt{1-\beta^2}} and c is the speed of light. Then, an observer...
  36. C

    Basis for Range of Linear transformation

    The problem is attached. The problem is "find a basis for the range of the linear transformation T." p(x) are polynomials of at most degree 3. R(T)={p''+p'+p(0) of atmost degree 2} This is pretty much as far as I got. I'm not sure how to do the rest. I'm thinking of picking a...
  37. C

    Finding basis for nullspace of transformation

    T: P2 → R (the 2 is supposed to be a subscript) The P is supposed to be some weird looking P denoting that it is a polynomial of degree 2. T (p(x)) = p(0) Find a basis for nullspace of linear transformation T.The answer is {x, x^2} I want to make sure I'm interpreting this correctly. It...
  38. P

    Linear Transformation in Mathematics

    I attached the problem, idk if it's really easy or If I'm doing it all wrong. Since T is a linear transformation T(u+v)=T(u)+T(v)=w+0=w?
  39. B

    Kaon in an accelerator, Lorentz transformation problem

    Homework Statement Ok so a Kaon (m = 500MeV) is accelerated from rest along the z-axis to a final energy of 5GeV, I need to find two factors of a lorrentz transformation β and γ and write a four vector for this. Homework Equations β=p/E γ=E/m The Attempt at a Solution I have...
  40. Mentz114

    Transformation of an acceleration vector under a basis change

    This thread is spawned from an earlier one https://www.physicsforums.com/showthread.php?t=647147&page=7 For the stationary ( ie comoving ) frame in the Schwarzschild spacetime the co-basis of the frame field is s_0= \sqrt{\frac{r-2m}{r}}dt,\ \ s_1=\sqrt{\frac{r}{r-2m}}\ dr,\ \ s_2=r\...
  41. L

    Help with Fibonacci Transformation

    Homework Statement A more efficient algorithm to calculate Fibonacci numbers applies the simultaneous transformation: T(a; b) = (a+b; a) repeatedly with a = 1 and b = 0 as initial values. What Fibonacci numbers result from T^k(1; 0)? Justify your answer (e.g., as proof by induction in...
  42. P

    Lorentz Transformation Rapidity

    Concerning Rapidity, if tanh(Fi) = v/c, can it be concluded in general that the relative angle of two frames in combination with Lorentz Transformation is tan(theta) = tanh(Fi) = v/c, where theta is the relative angle?
  43. N

    Inner Product as a Transformation

    Homework Statement Let V be an inner product space. For v ∈ V fixed, show that T(u) =< v, u > is a linear operator on V . Homework Equations The Attempt at a Solution First to show it is a linear operator, you show that T(u+g)=T(u)+T(g) and T(ku)=kT(u) So, T(u+g)=<v...
  44. N

    Linear Algebra - proof of transformation

    Homework Statement Suppose T: V -> W is linear. Prove that T(0) = 0 The Attempt at a Solution T(v) = Av T(0) = A(0) = 0 Is that right?
  45. H

    Lorentz Transformation: Proving θμ Covariant Vector

    Homework Statement Given that the derivative θμ transforms as a covariant vector ,show that θμ transforms as a contravariant vector. Homework Equations Please look the attachement The Attempt at a Solution Does anyone know how i should go to prove it ?Is it just a trivial...
  46. M

    Source Transformation: Power Simplified

    Is the answer correct?
  47. soothsayer

    What is the Correct Setup for a Lorentz Transformation Matrix?

    Homework Statement Show that the following is a Lorentz Transform: \Lambda _{j}^{i}=\delta _{j}^{i}+v^iv_j\frac{\gamma -1}{v^2} \Lambda _{j}^{0}=\gamma v_j , \Lambda _{0}^{0}=\gamma , \Lambda _{0}^{i}=\gamma v^i where v^2 =\vec{v}\cdot \vec{v}, and \delta _{j}^{i} is the Kronecker Delta...
  48. A

    Understanding the Concept of Canonical Transformation in Hamiltonian Mechanics

    There's a part in my book that I don't understand. I have attached the part and it is basically about how to transform from a set of conjugate variables (q,p) to another (Q,P) while preserving the hamilton equations of motion. I don't understand what he means by q,Q being separately independent...
  49. H

    Exploring a Conformal Transformation Between 2-D Space-Times

    I have two 2-dimensional space-times. One of them is flat the other one has not-vanishing curvature (Riemann tensor). But they seem to have a similar global and causal structure. Of course, because of the 2-dimensional case they are local conformally flat. I am looking for a relation between...
  50. G

    Why is dx/dt*(\delta x*dx/dt) equal to 1/2\delta(dx/dt)^2?

    Hello. I don't understand one transformation that is made on page 25 of this paper: http://www.atm.ox.ac.uk/user/read/mechanics/LA-notes.pdf It is the second equation from the top, ont the one marked as '2', but just the second one. dx/dt*(\delta x*dx/dt)=1/2\delta(dx/dt)^2 Why...
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