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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 ...
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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 ...
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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 $(\epsilon , \mu)$ ...
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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: $$c_1 \cdot f(g(x+c_2)) = ... 1answer 48 views transformations of \mathbb R^2 Consider the transformation (u,v)=f(x,y)=(x-y,xy). Demonstrate the effect of this transformation on the lines x-y=\text{constant}, x+y=0, and the curves xy=\text{constant}. In particular ... 3answers 126 views How to Evaluate \int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy How do you get from$$\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?$$I have tried using a change of variables formula but to no avail. Edit: ... 1answer 61 views Jacobian of an inverse Suppose that we have an invertible map T(u,v)=(x,y). The Jacobian of T is given by  \text{Jac}(T)= \left| {\begin{array}{cc} x_u & x_v \\ y_u & y_v \\ \end{array} } ... 2answers 105 views Laplace Transformation Applications In one of our Mathematics lecture our Prof told us that similar to Logarithmic Transformations we can use Laplace Transformations to solve difficult equations. What kind of equations do Laplace ... 1answer 243 views Which matrix transforms my vector field F(r,\theta,\phi) from cylindrical to spherical coordinates I am looking for the matrix that I have to apply my vector at the position (r,\theta, z) to in order to get the appropriate vector in spherical coordinates. I am totally okay, if you could give me ... 1answer 97 views Evaluate the following integral by transformation: 1 1-x ∫ ∫ (sqrt(x+y)(y-2x)^2)dydx 0 0$$ \int_0^1 \int_0^{1-x} \sqrt{x+y} \, (y-2x)^2 \,dy \, dx $$I've determined that u = x+y and v = y-2x and that the jacobian is = 1/3. and that x ... 0answers 55 views Stabilize Variance for Statistics (Transformation) Problem: When Y (> 0) has mean and variance equal to \mu and \mu/n respectively, it is shown in the textbook that the appropriate transformation of Y to stabilize variance is the square root ... 1answer 40 views Need to find a Fourier Series… I am to find a Fourier Series for the following function:$$ y(x)=\sqrt {R^{2}-x^{2}} $$about$$ -R \leq x \leq R $$with the recursion$$ y(x+2R)=y(x) $$Do I let\sqrt {R^{2}-x^{2}}equal y ... 0answers 34 views Identity for reducing the power in the parameters of Hypergeometric functions Is there any identity/formula for reducing/increasing the power in the parameters of the Gauss Hypergeometric function  _2F_1(a,b;c;z^d) (d is a real) let's say to z? Is there also any identity for ... 1answer 126 views How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation Let T be the triangel with vetrices ( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) . Evaluate the integral :$$ \iint_D e^{\frac{y-x}{y+x}} $$a) by transforming to polar coordinates b) by using the ... 1answer 82 views Transformations and coordinate Systems I am working on some practice exercises (not homework) on transformations and need some intuition and help. One of the questions is: (u,v)=f(x,y) where  \quad u= { e }^{ x }\cos(y), \quad v = { e ... 1answer 78 views Help in finding Jacobian I have$$\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} $$I know how to compute the ... 1answer 69 views Is there any sensible way to simplify this pde? Problem: Try to simplify$$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 ...
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 ...