0
votes
1answer
25 views

Finding the image of a region transformed by a mapping

The only examples I've found are either very complicated, or state the transformation like y=g(u,v) x=f(u,v), whereas this question states u and v in terms of x and y. I'm not sure how to get ...
1
vote
1answer
23 views

Transformation shifts parallelogram to trapezoid - fairly simple

We are given the region $D= {\{(x,y) | 1 \leq x-y \leq 2, x \leq 0, y\leq 0\} \subseteq \mathbb R^2}$ I drew this region on a piece of paper, it resembles an infinite parallelogram on the third ...
1
vote
1answer
37 views

Jacobian of a transformation in cylindrical coordinates

In an area called transformation optics, they transform Maxwell equations from one space coordinate system to another, and then somehow obtain the properties of background material $(\epsilon , \mu)$ ...
0
votes
0answers
12 views

When is $c_1 \cdot f(g(x+c_2)) = f'(x)g(x)$?

We are allowed to pick and $c_1, c_2$ that helps make this question easier. So when is $$c_1 \cdot f(g(x+c_2)) = f'(x)g(x) \tag{1}$$ Also, separately, I'm wondering: $$c_1 \cdot f(g(x+c_2)) = ...
1
vote
1answer
48 views

transformations of $\mathbb R^2$

Consider the transformation $(u,v)=f(x,y)=(x-y,xy)$. Demonstrate the effect of this transformation on the lines $x-y=\text{constant}$, $x+y=0$, and the curves $xy=\text{constant}$. In particular ...
2
votes
3answers
126 views

How to Evaluate $\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$

How do you get from$$\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$$to $$\frac{1}{2}\int^\infty_0\int^u_{-u}e^{-u^2} dv\ du?$$ I have tried using a change of variables formula but to no avail. Edit: ...
0
votes
1answer
61 views

Jacobian of an inverse

Suppose that we have an invertible map $T(u,v)=(x,y)$. The Jacobian of $T$ is given by $ \text{Jac}(T)= \left| {\begin{array}{cc} x_u & x_v \\ y_u & y_v \\ \end{array} } ...
1
vote
2answers
105 views

Laplace Transformation Applications

In one of our Mathematics lecture our Prof told us that similar to Logarithmic Transformations we can use Laplace Transformations to solve difficult equations. What kind of equations do Laplace ...
1
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1answer
243 views

Which matrix transforms my vector field $F(r,\theta,\phi)$ from cylindrical to spherical coordinates

I am looking for the matrix that I have to apply my vector at the position $(r,\theta, z)$ to in order to get the appropriate vector in spherical coordinates. I am totally okay, if you could give me ...
2
votes
1answer
97 views

Evaluate the following integral by transformation:

1 1-x ∫ ∫ (sqrt(x+y)(y-2x)^2)dydx 0 0 $$ \int_0^1 \int_0^{1-x} \sqrt{x+y} \, (y-2x)^2 \,dy \, dx $$ I've determined that $u = x+y$ and $v = y-2x$ and that the jacobian is $= 1/3$. and that $x ...
2
votes
0answers
55 views

Stabilize Variance for Statistics (Transformation)

Problem: When $Y (> 0)$ has mean and variance equal to $\mu$ and $\mu/n$ respectively, it is shown in the textbook that the appropriate transformation of Y to stabilize variance is the square root ...
2
votes
1answer
40 views

Need to find a Fourier Series…

I am to find a Fourier Series for the following function: $$ y(x)=\sqrt {R^{2}-x^{2}} $$ about $$ -R \leq x \leq R $$ with the recursion $$ y(x+2R)=y(x) $$ Do I let$\sqrt {R^{2}-x^{2}}$equal $y$ ...
1
vote
0answers
34 views

Identity for reducing the power in the parameters of Hypergeometric functions

Is there any identity/formula for reducing/increasing the power in the parameters of the Gauss Hypergeometric function $ _2F_1(a,b;c;z^d)$ (d is a real) let's say to z? Is there also any identity for ...
1
vote
1answer
126 views

How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral : $$ \iint_D e^{\frac{y-x}{y+x}} $$ a) by transforming to polar coordinates b) by using the ...
2
votes
1answer
82 views

Transformations and coordinate Systems

I am working on some practice exercises (not homework) on transformations and need some intuition and help. One of the questions is: $(u,v)=f(x,y)$ where $ \quad u= { e }^{ x }\cos(y), \quad v = { e ...
1
vote
1answer
78 views

Help in finding Jacobian

I have $$\begin{aligned}x_{1}&=r\sin(\theta_{1}),\\ x_{2}&=r\cos(\theta_{1})\sin(\theta_{2})\\ x_{3}&=r\cos(\theta_{1})\cos(\theta_{2}). \end{aligned} $$ I know how to compute the ...
0
votes
1answer
69 views

Is there any sensible way to simplify this pde?

Problem: Try to simplify $$x^2\frac{\partial^2w}{\partial x^2}+y^2\frac{\partial^2w}{\partial y^2}+z^2\frac{\partial^2w}{\partial z^2}+yz\frac{\partial^2w}{\partial y\partial ...
5
votes
1answer
482 views

A limit and a coordinate trigonometric transformation of the interior points of a square into the interior points of a triangle

The coordinate transformation (due to Beukers, Calabi and Kolk) $$x=\frac{\sin u}{\cos v}$$ $$y=\frac{\sin v}{\cos u}$$ transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the ...
1
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2answers
4k views

Non-Linear Transformation

Can someone explain to me in simple terms what a non-linear transformation is in maths? I know some single-variable calculus, but I read it has to do with multi-variable calculus, which I'm not ...