Transformation has many meanings in mathematics. If using this tag, add another tag related to the object being transformed. If there is a tag for your specific kind of transformations, use that one instead: e.g., (laplace-transform), (fourier-analysis), (z-transform), (integral-transforms), (rigid-...

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1answer
19 views

Interchanging vectors coordinates

Is there any relation between two vectors with interchanging coordinates .. i.e: the x component of the first is the y component of the second and vice versa.
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Proportinal line elements imply preservations of angles.

Consider a Riemannian manifold $(M,g)$, and a variation of the line element $\delta ds^2$ that is proportional to the original line element $ds^2$. This is $\delta ds^2=c ds^2$ for some constant $c$. ...
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1answer
29 views

How to rotate a coordinate system in $\mathbb{R}^3$ through an angle about an arbitrary axis passing through origin?

The question spurred in my mind when I was asked the following: Find the transformation matrix T that describes a rotation by $120^\circ$ about an axis from the origin through the point $(1,1,1)$....
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0answers
18 views

Mappings and directional derivatives

In one book on Complex Variables, it is said that if the function $h(u,v) = v+2$, the transformation $w=iz^2=i(x+iy)^2=-2xy+i(x^2-y^2)$ is conformal when $z\ne 0$. It maps $y=x$ (for $x>0$) onto ...
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1answer
17 views

Feature transformation

Does someone know a computationally efficient bijective function $f$ : $\mathbb{R}\rightarrow (y_{0},y_{1})$ ? Preferably, $(y_{0},y_{1})=(-1,1)$ and $(0,1)$.
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31 views

Trigonometric identity (Backlund permutability theorem)

I have been studying Backlund transformations using Rogers and Schief, and I am now reading about the permutability theorem. I understood everything up to the very last part for the permutability. It ...
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21 views

Proof that Kronecker Delta is Mixed Tensor

The book I am reading asks the reader to verify that the Kronecker Delta is a second-order mixed tensor with one contravariant and one covariant index as indicated: $$ \delta_j^i = \left\{\begin{...
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1answer
36 views

Understanding how to change coordinate systems

Let's say I have a canonical coordinate system in $\mathbb{R}^2$ described by the basis $\{e_1=\begin{bmatrix}0\\1\end{bmatrix}, e_2=\begin{bmatrix}1\\0\end{bmatrix}\}$. In it I have a vector $\vec{x}...
2
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1answer
68 views

How do I draw this picture in squares of discrete $\sqrt{z}$?

From Richard Kenyon's homepage gallery: I want to understand the mathematics of this, and similar/related transformations. ... An explanation in words (1st year uni level maths) would be ideal. I'...
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0answers
9 views

Probability density transformation for non-invertible mapping

I am looking for a generalization of the result which states that the density of the sum of two random variables is the convolution of their densities. Specifically, if I have $Z=f(X,Y)$, where $p_{X,...
2
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1answer
23 views

matrix transformation of deformed rectangle

I am working on touch screen calibration, and have come across a problem. My area of touch screen input is a Trapezoid which looks like a square on one side and a triangle on the other. (the angle ...
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2answers
45 views

translation and rotation of a parabola

I am trying to translate a parabola to the origin, rotate by T radians and then translate back to the original position. I can calculate the new X and Y vectors using matrix operations and the regress ...
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1answer
54 views

Looking for the name of a formula [closed]

Probability subject. The question is: If fx(x)=xe^(-x^2/2) for x>0 and Y=lnX find the density function for Y The solution is:(e)^(2y-1/2e^2y) I'm stuck on the part of the solution that uses this ...
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0answers
27 views

Proving non-linear mapping is invertible using partial derivatives only

Given $f : \mathbb{R} \rightarrow \mathbb{R}$, it's possible to show that $f$ is a bijection by considering its derivatives only: if the derivative is always positive or always negative, then the ...
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1answer
50 views

What is the domain for which this integral transform is defined?

Let $s=\sigma+it$ the complex variable where thus $i^2 =-1$, and $\sigma$ and $t$ are real numbers. Let $\mu(k)$ the Möbius function. It is possible determine the set of functions such that $$M \...
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1answer
26 views

Problems again with an isomorphism

Let $X$ and $Y$ be arbitrary sets and $f:X\rightarrow Y$ an isomorphism. Prove that there exist a transformation $g:Y\rightarrow X$ such that $f\circ g$ is the identity in $Y$. I can't start the ...
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1answer
74 views

Active and passive transformations in Linear Algebra

I am trying to understand what each transformation means and what their differences are but many books that don't state which transformation they are referring to make it a bit confusing to understand ...
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4answers
66 views

The formula for 3D rotation of the perspective of an image in 2D space

Do you know what the formula is for the 3D rotation of the perspective of a 3D object in a 2D space. For example, imagine that we got a picture of a 3D object. So, we have the projected picture of ...
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0answers
24 views

rotating a point using a previously rotated one

I want to rotate a shape in an n dimensional space (n>3) around (about) the origin. knowing the outcome of rotation on a point like A, which is A', how can I find the rotation outcome on a point like ...
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1answer
31 views

Find the marginal distribution of $V=X-Y$

Problem: Show that the marginal density function of $f_V(v)$ if $V=X-Y $ is $$f_{V}(v)= \frac{1}{(1+|v|)^2}$$ for $ -\infty < v < \infty $. When the bivariate density function $f_{X,Y}(x,y)$ is ...
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3answers
42 views

How to tansform ${\sqrt{n-1}} + {\sqrt{n+1}} = q$ into $q^4 - 4q^2n + 4 = 0$? [closed]

Please help me tansform ${\sqrt{n-1}} + {\sqrt{n+1}} = q$ into $q^4 - 4q^2n + 4 = 0$?
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0answers
19 views

Gaussian function at a rotated and translated coordinate system

I'm reading a paper and this coordinate transformation came along. In the $z_{i}=0$ plane the electric field is writen as $E=\exp[-x_{i}^2]$. The author says it's more convenient to work with the ...
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2answers
27 views

Transformation of a plane

I have the $(x,y)$-plane $$\left\{(x,y,z)\in \mathbb{R}^3 | x,y\in \mathbb{R}, z = 0 \right\}.$$ I need a transformation matrix to transform this to the plane $$ \left\{ (x,y,z) \in \mathbb{R}^3 | x+...
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1answer
34 views

Mathematics of transformation of 2-D to 1-D coordination.

Let's see an example In cartesian coordinate system: ...
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0answers
23 views

Transformation of the gradient

For a function $f\in C^2$, $f:\mathbb{R}^n\to\mathbb{R}$ and a point $x\in\mathbb{R}^n$ with $\nabla^2f(x)$ positive definit one can calculate the new point $x^+=x+s$ as follows: Change the ...
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1answer
28 views

How can I calculate the angle of a line/vector if the center of the image is not (0,0)?

Simple image about the problem How can I calculate the alpha? My center of the image is (320,240) because it is a 640x480 image and the upper left corner is the (0,0). I tried to calculate it with ...
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2answers
40 views

Finding the function of a sine graph that has both translation and transformation

I can't quite find a problem similar enough to this yet, and I need some serious help. Here is a photo of the graph of the function I am trying to find out: Sorry, but I don't have enough ...
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1answer
61 views

2D Fourier transform of characteristic function of stripe on xy plane

Given a stripe $X$ on the xy-plane, namely $X\subset\mathbb{R}^2$, with $X=\{(x,y)\,|\; mx-\frac{1}{2}t \le y \le mx + \frac{1}{2}t$} and its "characteristic" function $$ f(x,y) = \begin{cases} 1, ...
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1answer
34 views

Find functions $F(\mathbf{x})$ invariant under a map $\mathbf{x} \to \mathbf{x'}$

We introduce a map $\mathbf{x} \to \mathbf{x'}$, defined as (for example on $\mathbb{R}^3$): $$x'=f(x,y,z) \\ y'=g(x,y,z) \\ z'=h(x,y,z)$$ Note that $f,g,h$ are not all linear (or at least, I'm not ...
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30 views

How to prove this identity? Transformation theorem

Let $A\in\mathbb R^n$ be a measurable set with finite measure. For a fixed vector $p\in\mathbb R^{n+1}$ define a cone with basis $A$ and peak $p$ as $$K(A,p)=\{tp+(1-t)q \in\mathbb R^{n+1} \,| \, q \...
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1answer
30 views

Coordinate transform a triangle

There is a triangle with points P1(x1,y1),P2(x2,y2),P3(x3,y3) on an XY plane. The final ...
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2answers
33 views

Transformation of a sphere and computing an integral by using sphere coordinates

Let $V \subset\mathbb R^3$be the ellipsoid $$9x^2+4y^2+z^2≤36.$$ How can I express $V$ as a transformation of a sphere and how can I compute the sphere $$\int_v x^2\,d\lambda^3(x,y,z)$$ with sphere ...
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1answer
48 views

Transformation matrix

For $x \in \mathbb{C}$ define $A,B \in M(3\times3, \mathbb{C})$ as $$ A = \begin{pmatrix} x & 0 & 0 \\ 0 & x & 1 \\ 0 & 0 & x \end{pmatrix}$$ and $$ B = \begin{pmatrix} x & ...
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2answers
43 views

Bilinear transformation which maps $z=(\infty, i, 0)$ and $w= (-1, -i, 1)$

I have three equations after simplifying this a bit $a+c=0$ $ai+b-c=0$ $b-d=0$ How do I proceed further? If you care to know this is from the chapter Complex Variables
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1answer
44 views

Is there a universal symbol for transformation or operation?

If I'm talking about a bunch of transformations (translation, rotation, scale, skew, etc) and I want to state $A$ [some transformation] $= B$, what would be the symbol for [some transformation] or ...
2
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2answers
50 views

What is the difference between and projection and a reflection, in vector transformation?

In my text book I have the problems of finding the standard matrix of the given linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$; Projection onto the line $y = -x$. Reflection in the line ...
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1answer
17 views

Derive the length of the longest line segment that can be enclosed inside the region A.

Q. Let A be the region in the xy-plane given by A={(x,y): x=u+v, y=v, u^2+v^2≤1}. Derive the length of the longest line segment that can be enclosed inside region A. My attempt: I found the equation ...
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1answer
26 views

Converting coordinates

I am having huge trouble converting between cartesian to polar and to spherical. I need these methods for my limits of integration in most cases but using the formulas never seem to work, I just ...
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1answer
30 views

Distribution of a transform of bivariate to univariate random variable?

Suppose we have two random variables $$R\sim U[1-\varepsilon,1]\;\;\;\;\; \Theta\sim U[0,2\pi],$$ and a third random variable $$X=g(R,\Theta)=R\cos\Theta.$$ What is the density $p_X(x)$? The ...
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1answer
17 views

inverse Mapping in Transformation of a random variable

I have a question concerning the the inverse mapping in the image . text extracted from Casella Statistical inference $g^{-1}(A) = \{ x \in \chi : g(x) \in A\}$ I know the idea that they want to ...
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0answers
22 views

Determinant in the transformation theorem

Where does the $|det|$ come from in the transformation theorem? It is pretty much the first time I saw a $|det|$ in analysis.
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19 views

Rotation of a coordinate system

Suppose that I rotate the (traditional) coordinate system $(x,y)$ by an angle $\theta$ to obtain a new coordinate system $(s,n)$. Consider a velocity vector $$v = (v_x,v_y) = v_xe_x + v_y e_y,$$ where ...
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1answer
27 views

Cone under similarity transformation

Suppose we have a cone passing through the origin of $xyz$ coordinate system. Now, the question is that whether we can find an invertible transformation on this coordinate system that turns the cone ...
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21 views

Is there a mathematical term for the “de-peaking” of a dataset

I have several sets of data that exhibit similar "peaks" when viewed as a graph, e.g.: A more useful representation of this data (for my purposes) is ${y = abs(x - 70)}$. The representation of the ...
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1answer
27 views

Probability - Finding the Support of a Joint Transformation

$$ f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad \text{elsewhere} \end{array} \right. $$ ...
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Find the basis $\mathcal{V}$ of $\mathbb{R}^4$ and $\mathcal{W}$ of $\mathbb{R}^3$.

Let $T:\mathbb{R}^4\to\mathbb{R}^3$ be a linear function with the transormation matrix given as: $$A=\begin{pmatrix} -3 & 2 & 3 & -3 \\ 4 & 0 & -4 & 4 \\ 2 & 0 & -2 &...
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1answer
46 views

If $f_X(x) = x/8$, find the pdf of $Z = \log (x/4)$.

Given the function $f_{X}(x) = \left\{ \begin{array}{lr} \ \frac{x}{8} \quad & : 0 <x <4 \\ 0 & :\text{Otherwise} \end{array} \right. $ Find the PDF of $Z = \log_{e}(\...
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1answer
26 views

Lineaire transformation of matrices, how to tackle

I've been learning linear algebra but can't understand the concepts of linear transformation. Correct me where i'm wrong: Say i'm given $T:R^2 \longrightarrow R^2$ is my transformation. This tells ...
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1answer
29 views

Co-ordinate transformation of metric

In a past exam paper that I am using to prepare for my upcoming finals, I have encountered the following question (paraphrased): Given the metric: $$\mathrm{d}s^{2} = -c^{2}\:\mathrm{d}t^{2}+\left(...
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3answers
49 views

How to graph $x^2 -4x$?

I know about transformations and how to graph a function like $f(x) = x^2 - 2$. We just shift the graph 2 units down. But in this case, there's an $-4x$ in which the $x$ complicated everything for me. ...