Questions tagged [change-of-variable]
This concern all problem requesting techniques and tricks about changes of variables in both computation of limits and integrals
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Why is $\int u\frac{dv}{dx} \, dx = \int u\ dv $? (Change of Variable while deriving Integration by Parts)
I have been learning the integration by parts formula.
$$\int u\ \mathrm dv = uv \ - \ \int v\ \mathrm du $$
I understand how the formula is derived when we keep everything in terms of $x$ (with $f(x)$...
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Does Fubini's theorem apply on this infinite region?
I came across the following example for a triple integral:
Find the volume of the region bounded by hyperbolic cylinders:
$$ xy = 1 \quad , \quad xy = 9$$
$$ xz = 4 \quad , \quad xz = 36 $$
$$ yz = 25 ...
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Constant change of variable of non-injective map
I have a specific integral
$$\int_{\mathbb{R}^3} f(\varphi(x))dx,$$
for an integrable function $f$ and some smooth function $\varphi$. I want to apply the change of variable $x\mapsto y=\varphi(x)$. ...
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Integration limits after multivariable change
I am making the following variable change from $x_i,y_i\to u_i,v_i,w_i$ where we define
$$u_1=x_1+x_2+y_1,\quad u_2=x_1+x_2-y_1,\quad u_3=x_1-x_2+y_1,\\
v_1=x_3+x_4+y_2,\quad v_2=x_3+x_4-y_2,\quad v_3=...
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Is it possible to make a posynomial concave using a change of variables?
Update
This question has an answer here.
Consider the following posynomial with respect to the variables $x_1,\dots,x_n$:
$$
\begin{align}
f(x_1,\dots,x_n) &= \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{...
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Change variable under partial derivation.
Hy all, i will take $x\in \mathbb R$ and $y\in\mathbb R^n$.
I wondering about what happen if i have $\partial_{x}u(e^{-x},y)$ and i do variable change $s=e^{-x}$. How is change $\partial_{x}u(e^{-x},y)...
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How to change integral bounds when calculating an improper integral
So it is estabilished that for definite integrals, the following holds:
$$\int_a^b f(g(x))g'(x)dx = \int_{g(a)}^{g(b)} f(u)du$$
given
$ u=g(x)$.
However, when calculating the expectation of a standard ...
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Convert Laplace's equation into polar coordinates
When I read Ahlfors's book, I came across this problem
the$$u(x,y)$$is a harmonic function
and satisfies Laplace's equation
$$\Delta u＝\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^...
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Equivalence with the Weierstrass transform
I have the following expression
$$\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{+\infty}dx~ f(x-y) e^{-x^2/4 t} \tag{1},~~\forall ~y \in \mathbb{R}.$$.
I am trying to relate it with the generalized ...
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Most general conditions for variable substitution with Riemann integral
This question is motivated by the discussion here:
https://matheducators.stackexchange.com/a/26687/117
Let $g$ be defined and differentiable on an interval containing $[a,b]$ and $f$ be defined on an ...
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Complicated change of variables formula with CDF of normal distribution instead of diffeomorphism
I have a parameter $\theta\in[0, 10]^4$ and a variable $z\in\mathbb{R}^m$. Consider the following integral
$$
I(\theta, z) := \int F(\theta, z) \mathbb{I}(\|f(\theta, z)\| \leq 1) p(\theta, z) dz d\...
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Is there a form for the inverse of $(1-l x) ^K(1+w x)^{(T-K)}$ in terms of "named" functions (Beta, Gamma, etc)?
Let $l \in (0,1)$, $w>0$, $K$ and $T$ positive integers.
The function $f(x) = (1-l x)^K(1+w x)^{(T-K)}$, restricted to $x \in [0,1]$ is then logconcave in x.
Thus, we can separate the domain into ...
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Unable to spot the mistake in change of variables in $\mathbb{R}^N$
Let $Q \subset \mathbb{R}^N$ be the cube of side length 2 centred on the origin, $Q_+, Q_0$ and $Q_-$ be the upper half, equatorial and lower halves of the cube. Let $u \in C(Q_+)$ and $\varphi \in C^{...
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Changing the Variable of Integration with Product of Functions
I am a bit confused about how to tackle the following integration and was hoping to find some help here.
Problem
I have an integral of the following form
$$
\int_{x_i}^{x_f}\! f(a(x),y)\cdot j_l\left(...
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Finding the area bounded by some curves using change of variables and double integrals
find the area of the first-quadrant region bounded by the curves $y = x^3,y=2x^3,x=y^3,x=4y^3$ be the curves that bound the area:
now we can use the substitution:
$$y =ux^3$$
$$x =vy^3$$
in order to ...
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Solve for $x$ and $y$ in terms of $u$ and $v$. Then compute the Jacobian.
solve for x and y in terms of u and v. Then compute the Jacobian
let: $u = x+y$ and $v = x-y$
since:
$$\frac{\partial (x,y)}{\partial(u,v)}* \frac{\partial (u,v)}{\partial(x,y)} = 1$$
but
$$\frac{\...
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How to write the distribution of the (Euclidean) distance between two random points with arbitrary distributions
Let the random variable $ \hat{{\mathbf x}}_i $ be defined over a some D-dimensional space (I'm interested in D=2), with distribution $ f_i(\mathbf x) $. What is the distribution of Euclidean distance ...
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Change of variables and fiber bundles
The random variable X has some density p(x) in Lebesgue measure and I know that X=f(A, B), for example X=A^2+B^2. How can I find the corresponding density over A, B if I assume it is distributed ...
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Convex Optimization: Is convexity of the constraint necessary for a correct analytical solution?
I am from the field of economics. For an agent's utility maximization it's common to use convex optimization following the lagrange maultiplier method. This is mostly used to get an analytical result; ...
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Conditional change of variables
Goal. I want to know if the proposition below is true when $\beta(x,y)$ is not differentiable in $x$. If the proposition is false, I'd like a counterexample.
Setup. Set $X=\mathbb{R}^m$ and $Y=\mathbb{...
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Effect of Increment of Variable on Output in a Multivariate Equation
There is an equation as you can see below. Here, C1, C2 and C3 denote constants, while x and y denote variables.
Equation
I want to observe the effect of x on U. In short, will increasing x increase ...
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Integral for the surface area of a half n-sphere
I am trying to evaluate the following integral on $\mathbb{R}^{n-1}$
$$\int_{\mathbb{R}^{n-1}}\frac{1}{(1+|x|^2)^{\frac{n}{2}}}dx$$
I claim that this is equal to the half the surface area of the ...
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Integration including the CDF of a standard normal distribution
I am working on a project, where the following integral shows up, and I don't see how to solve this, or to show if it has a closed form solution.
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Phi(z)z ...
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How to understand change of variables intuitively?
I've been trying to prove or have an intuitive understanding of the change of variables, and I tried it for the function $f(x)=x^2$ using $u(x)=x^2$, the transformed function then becomes $g(u)=u$.
...
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Finding double integral over balls
I am facing trouble in evaluating some double integral. The definition of Riesz energy is given by $I_t(U)=\int_U\int_U|x-y|^{-t}\ dx\ dy$ where $U$ is an open subset. For better understanding I want ...
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Variable changes in matrix integrals
I need to evaluate the following matrix integral:
$I = \int \mathrm{etr}\left(-cX^TX\right)\mathrm{det}(I_d + AX^TBX)^{-k/2}dX$
where $A$, $B$, and $X$ are $d\times d$, $A$ and $B$ are positive ...
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Help with a variable change
I'm integrating a function of the variables $x_i,y_i,z_i$ for $i=1,2,3$ and I define new variables given by
$$u_1=x_1+y_3 z_3,$$
$$u_2=x_2+y_1 z_1,$$
$$u_3=x_3+y_2 z_2.$$
NOTE: I don't actually want ...
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Given $\frac{\partial^2u}{\partial t^2}=\frac{\partial^2u}{\partial x^2}$, $\xi=x+t,\eta=x-t$, show $\frac{\partial^2 u}{\partial\xi\partial\eta}=0$ [duplicate]
Given $u(x,t)$ that satisfies the following wave equation for all $x,t$:
\begin{align*}
\frac{\partial^2 u}{\partial t^2} &= \frac{\partial^2 u}{\partial x^2}
\end{align*}
Let $\xi = x + t$, $\...
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Change of variable in multivariate integral, from one variable to two
I remember I learned how to do the inverse of this, but I'm looking two things, one is the right way for the next change of variable, second the demonstration.
Z function give the third axis in ...
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Follow-up question on Abel Summation
This is a follow-up to a much simpler question I asked here, which @PrincessEev answered promptly and perfectly. She showed me how to rewrite the sum $\sum _{i=1}^x \phi (x-i)$ in such a way that Abel ...
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Can I write an Abel Summation Formula for this?
Given appropriate constraints, and a continuously differentiable real-valued function $\phi (x)$, the Abel Summation Formula (Wikipedia article here) can be written as
$$\sum _{k=1}^x \phi (k) = \...
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Jacobian change of variable leading to strange answer
This is an extension of another question.
Let $X_1$ and $X_2$ be independent exponential random variables such that
$$
f(x_1) =
\begin{cases}
\frac{1}{\beta}e^{-\frac{x_1}{\beta}}, & x>0 \\
\...
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What is Meant by "Change of Origin" in Coordinate Geometry?
I don't think I understand what is meant by "to shift the origin of coordinates to the point $(h,k)$ in coordinate geometry. I've read Loney's book on coordinate geometry in which he says that to ...
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Change of variable when mapping to a lower dimension
A similar question has been asked before in this link, however the first answer does not address the point and considers an approach which is not related to the question, in my opinion.
To formulate ...
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Change of variables in Oppenheim example 2.5
How do they do the change of variables from $$\sum_{k=-\infty}^{n} \rightarrow \sum_{l=-n}^{\infty}$$
For the top, you can write $n = m-l = m + k$, so as $k \rightarrow -\infty$ then $n\rightarrow -\...
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joint density of a random variable and its bijective, differentiable function
Consider a random variable $X$ and its density function $f_{X}(x)$, consider a bijective, differentiable function $H$, and let the random variable $Y=H(X)$. I am trying to compute the joint density ...
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Change of variable under surface integrals
I'm seeking clarification on the following identity involving surface integrals and partial derivatives:
$B_{\rho} = B(y, \rho) \subset \mathbb{R}^n$ represents the ball centered at $y$ with radius $\...
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How to find new bounds when doing this change of variable on double integral
The region of integration looks like as follows:
I started by computing the jacobian which I found to be $|J(\rho,\theta)|=\frac{-e^{-2\rho}}{6}$. And with the new variables the integrand becomes $\...
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Change of Variables over an $n$-dimensional ball
I wish to understand what is going on in this change of variables, specifically what the transformation function and Jacobian are. The result is as follows, if we are integrating a function over a an $...
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Change of variable formula for multivariate integral with constrain
Suppose $g: \mathbb{R}^{n} \to \mathbb{R}$ is $C^{1}$ and satisfies $\nabla g(x) \neq 0$ for every $x \in \mathbb{R}^{n}$. Let $E \in \mathbb{R}$ be fixed. I want to find an explicit expression for ...
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Analysing properties of a function in integral form
let:
$$F(x) = \int_0^x\sqrt{t}\sin(t)dt$$
Say whether $F(x)$ is positive, negative or zero at each of the following points, and give a reason in each case:
$$x = \pi$$
$$x = 2\pi$$
I was trying:
$$F'(...
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Chain rule (multivariate) and implicit differentiation problem
I have been worked out a solution for the following problem, but I am wondering if there is an easier way to solve it. I would be very grateful for any suggestions!
Let $z=f(x,y)$ a function of two ...
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change of variable in multivariable calculus [closed]
I am taking $x=r\cos\phi, y=r \sin\phi$
$ dx=-r \sin\phi d\phi+\cos\phi dr, dy=r\cos\phi d\phi + \sin\phi dr$
it is working with jacobian of$(x,y) $concerning $(u,v)$ and cross product i.e. if we ...
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2
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Evaluate $\iint_D|x - y|\mathrm{d}x\mathrm{d}y$ where $D = \{(x,y)\in\mathbb{R}^{2} \mid x^{2} + y^{2} \leq 2(x + y)\}$
I have problem with solving the following integral.
Given that $D = \{(x,y)| x^2 + y^2 \leq 2(x+y)\}$, evaluate the double integral:
$$\iint_D|x - y|\mathrm{d}x\mathrm{d}y$$
I think I should convert $...
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Derivation of expectation of random variable with delayed differential equation
I have problem deriving the expectation of random variable from a delayed differential equation. Any comment or advice for getting the result would be much appreciated!
The distribution of random ...
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How would the bounds of integral change under matrix multiplication and shift?
I want to do an integral ${x}$ over multidimensional Gaussian $e^{-|y|^2}$ where $y = LeakyRelu(Mx + b)$. Eventually, I want to do this integral after multiple transformations (i.e. $y_{i+1}= ...
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Frobenius method in differentials equations
I'm having problems solving this exercise:
Prove that, if $x(t)=t^\alpha y(t), \alpha\in\mathbb{R}$ satisfies:
$$0=t^2x''(t)+tp(t)x'(t)+q(t)x(t) \quad \forall t \in (0,r) $$
where $r>0$ is the ...
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Probability distribution of speed at a given time vs probability distribution of speed at a given position
Here is a cute problem; I am looking for a straightforward proof.
Consider you are in a car that drives on a one-way street, and its speed is a random function of time. You are given a probability ...
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How to change index in summation
Pease help me understand how they have changed index of summation from r to n here. If we take $$n = r-s$$ how n is changing from -$\infty $ to $\infty$
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Change of Variables for Parametrized Manifolds
Suppose $M=\varphi(A)$ a parametrized $k$-manifold in $\mathbb R^n$, given a diffeomorphism $g$ in $\mathbb R^n$, $N=g(M)$ is still a parametrized $k$-manifold. Now for a scalar function $f:N\to\...
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