# Tagged Questions

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-...

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.
6 views

### 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 $c$. ...
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 $(1,1,1)$....
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 ...
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 $(0,1)$.
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 ...
21 views

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 ...
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 ...
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 ...
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 ...
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 ...
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 $q^4 - 4q^2n + 4 = 0$?
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 ...
27 views

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 ...
30 views

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
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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 ...
27 views

### Probability - Finding the Support of a Joint Transformation

$$f(x,y) = \left\{ \begin{array}{ll} 12xy(1-y) & \quad 0< x < 1, 0<y<1 \\ 0 & \quad \text{elsewhere} \end{array} \right.$$ ...
### 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. ...