What is Change of variables: Definition and 219 Discussions
In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem.
Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation (chain rule) or integration (integration by substitution).
A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial:
x
6
−
9
x
3
+
8
=
0.
{\displaystyle x^{6}-9x^{3}+8=0.}
Sixth-degree polynomial equations are generally impossible to solve in terms of radicals (see Abel–Ruffini theorem). This particular equation, however, may be written
(
x
3
)
2
−
9
(
x
3
)
+
8
=
0
{\displaystyle (x^{3})^{2}-9(x^{3})+8=0}
(this is a simple case of a polynomial decomposition). Thus the equation may be simplified by defining a new variable
u
=
x
3
{\displaystyle u=x^{3}}
. Substituting x by
u
3
{\displaystyle {\sqrt[{3}]{u}}}
into the polynomial gives
u
2
−
9
u
+
8
=
0
,
{\displaystyle u^{2}-9u+8=0,}
which is just a quadratic equation with the two solutions:
u
=
1
and
u
=
8.
{\displaystyle u=1\quad {\text{and}}\quad u=8.}
The solutions in terms of the original variable are obtained by substituting x3 back in for u, which gives
x
3
=
1
and
x
3
=
8.
{\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.}
Then, assuming that one is interested only in real solutions, the solutions of the original equation are
$$h(t)=f(t)*g(t)=\int_0^t f(\tau)g(t-\tau)d\tau=\int_0^t g(\tau)f(t-\tau)d\tau\tag{1}$$
The Laplace transform is
$$H(s)=\int_0^\infty h(t)e^{-st}dt=\int_0^\infty\left ( \int_0^t g(\tau)f(t-\tau)d\tau\right )e^{-st}dt\tag{2}$$
The Laplace transforms of $f$ and $g$ are
$$F(s)=\int_0^\infty...
In Greiner's Classical Electromagnetism book (page 126) he has a derivation equivalent to the following.
$$\int_V d^3r^{'} \nabla \int_V d^3r^{''}\frac {f(\bf r^{''})}{|\bf r + \bf r^{'}- \bf r^{''}|}$$
$$ \bf z = \bf r^{''} - \bf r^{'} $$
$$\int_V d^3r^{'} \nabla \int_V d^3z \frac {f(\bf z +...
Hey all,
I am currently struggling with a change of variables step in my calculations.
Suppose the solutions ##f_{1}(x)## and ##f_{2}(x)## of the following system of differential equations is known:
Now the system I wish to solve is:
Upon first glance, it seems that the association ##f_{2}(-x)...
Greetings all.
I just got confused by the following.
Consider volume integral, for simplicity in 1D.
$$
V(A) = \int_{A} dz.
$$
If ##z## can be written as an invertible function of ##x##, i.e. ##z=f(x)##, we know the change of variables formula
$$
V(A)=\int_{A} dz= \int_{z^{-1}(A)} |z'(x)|dx...
R is the triangle which area is enclosed by the line x=2, y=0 and y=x.
Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v##
## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...
Solution 1:
The answer is ## \frac{2b^5\pi}{5} \times \left(1 -\frac{a}{\sqrt{1+a^2}}\right)##
Solution 2:
I want to decide which answer is correct? Would you help me in this task?
Find the volume V inside both the sphere $x^2 + y^2 + z^2 =1$ and cone $z = \sqrt{x^2 + y^2}$
My attempt: I graphed the cone inside the sphere as follows. But I don't understand how to use the change of variables technique here to find the required volume. My answer without using integrals is...
Summary: Find the volume V of the solid inside both ## x^2 + y^2 + z^2 =4## and ## x^2 +y^2 =1##
My attempt to answer this question: given ## x^2 + y^2 +z^2 =4; x^2 + y^2 =1 \therefore z^2 =3 \Rightarrow z=\sqrt{3}##
## \displaystyle\iiint\limits_R 1dV =...
Summary: Evaluate ##\displaystyle\iint\limits_R e^{\frac{x-y}{x+y}} dA ## where ##R {(x,y): x \geq 0, y \geq 0, x+y \leq 1}##
Author has given the answer to this question as ## \frac{e^2 -1}{4e} =0.587600596824 ## But hp 50g pc emulator gave the answer after more than 11 minutes of time...
Hello,
This problem comes just prior to introducing change of variables with Jacobian.
Given the following region in the x-y plane, I have to choose (with justification) the correct change of variables associated, for ##u\in [0,2]## and ##v \in [0,1]##.
The correct choice here is a), but I do...
I have the following differential equation, which is the general Sturm-Liouville problem,
$$
\dfrac{d}{dx} \left[ p(x) \dfrac{d\varphi}{dx} \right] + \left[ \lambda w(x) - q(x) \right] \varphi(x) = 0\ ,
$$
and I want to perform the change of variable
$$
x \rightarrow y = \int_a^x \sqrt{\lambda...
I am refreshing on the pde's, and i am trying to understand how the textbook was addressing change of variables, i find it a bit confusing. I will share the textbook approach, then later share my own understanding on change of variables approach. Here is the textbook approach;
My approach on...
I have a question about changing variables in the context of thermodynamics, but I suppose this would extend to any set of variables that have defined and nonzero partial derivatives on a given set of points. First I should define the variables.
##T## is temperature, ##U## is internal energy...
Hi,
This is as part of a larger probability change of variables question, but it was this part that was giving me problems.
Question: If we have ## 0 < x_1 < \infty ## and ## 0 < x_2 < \infty ## and the transformations: ## y_1 = x_1 - x_2 ## and ## y_2 = x_1 + x_2 ##, find inequalities for...
We have two different integrals, the first one being ∫∫erdrdθ where -1≤r≤1 and 0≤θ≤π which equals approximately 7 and ∫∫erdrdθ where 0≤r≤1 and 0≤θ≤2π which equals approximately 11. Why do these integrals have different values and do not go against the change of variables theorem?
I'm having...
Hi,
I was attempting the problem above and got stuck along the way.
Problem:
Suppose that ## Y_1 ## and ## Y_2 ## are random variables with joint pdf:
f_{y_1, y_2} (y_1, y_2) = 8y_1 y_2 for ## 0 < y_1 < y_2 < 1 ## and 0 otherwise. Let ## U_1 = Y_1/Y_2 ##. Find the probability distribution ##...
I've been trying to get change of variables in PDEs down (I don't particularly like my textbook or professor's approach to it), and I want to ask here if I am getting this right. Let ##\vec{x}=(x_1,x_2,...,x_n)^T## and ##\partial_\vec{x}=(\partial_{x_1},\partial_{x_2},...,\partial_{x_n})^T##. I...
I have a problem where I am given the density of states for a Fermion gas in terms of momentum: ##D(p)dp##. I need to express it in terms of the energy of the energy levels, ##D(\varepsilon)d\varepsilon##, knowing that the gas is relativistic and thus ##\varepsilon=cp##.
Replacing ##p## by...
Summary:: Calculate a double integral via appropriate change of variables in R^2
Suppose I have f(x,y)=sqrt(y^12 + 1). I need to integrate y from (x)^(1/11) to 1 and x from 0 to 1. The inner integral is in y and outer in x. How do I calculate integration(f(x,y)dxdy) ?
My Approach: I know that...
I'm guessing that there must be some nuance that I do not quite understand in the difference between ##|p\rangle## and ##|E\rangle##?
Like, later in the book even ##dk## is used as a variable of integration, where ##k = p/\hbar.## Surely this has effects on the value of the integral - wouldn't...
So we have ##x=\beta(1/2 mv^2-\mu)##, i.e ##\sqrt{2(x/\beta+\mu)/m}=v##.
##dv= \sqrt{2/m}dx/\sqrt{2(x/\beta+\mu)/m}##.
So should I get in the second integral ##(x+\beta \mu)^{1/2}##, since we have: $$v^2 dv = (2(x/\beta+\mu)/m)\sqrt{2/m} dx/\sqrt{2(x/\beta+\mu)/m}$$
So shouldn't it be a power...
Hello-
In the attached screenshot from my textbook, I am trying to understand how they get from equation 6.5 to 6.5a. I have attached my attempt to solve it, but I am stuck evaluating the left side. I do not see how to get their result.
Relevant information:
k, T_w, T_inf, h and L are all...
Hi,
I understand the underlying concept of changing variables in PDEs (so that we can reduce it to a simpler form), however, I am just not completely clear on the mathematics of it so I have a quick question about it.
For example, if we have the transmission line equation \frac{\partial...
Summary: When ##V (x) = \frac 1 2 mω^2x^2 + mgx##
##H=\frac p 2m +V(x)##
Difficulty understanding how these change on variables came about
##y = x+\frac mg mω^2 = x+\frac g ω^2##
Apologies if this is not the appropriate thread. I chose this one because even though it's physics, I'm having...
Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##.
___________________________________________________________________________
Consider the following multiple integral:
##\displaystyle B= \iint_S \Biggl( \iiint_{V'}...
Homework Statement
Given that ##x=\phi (t)##, ##y=\psi(t)## is a solution to the autonomous system ##\frac{dx}{dt}=F(x,y)##, ##\frac{dy}{dt}=G(x,y)## for ##\alpha < t < \beta##, show that
##x=\Phi(t)=\phi(t-s)##, ##y=\Psi(t)=\psi(t-s)##
is a solution for ##\alpha+s<t<\beta+s## for any real...
Homework Statement
The assignment is to transform the following differential equation: ##x^2\frac {\partial^2 z} {\partial x^2}-2xy\frac {\partial^2 z} {\partial x\partial y}+y^2\frac {\partial^2 z} {\partial y^2}=0##
by changing the variables: ##u=xy~~~~~~y=\frac 1 v##Homework Equations...
In transforming an integral to new coordinates, we multiply the “volume” element by the absolute value of the Jacobian determinant.
But in the one dimensional case (where “change of variables” is just “substitution”) we do not take the absolute value of the derivative, we just take the...
Homework Statement
Let D be the triangle with vertices (0,0), (1,0) and (0,1). Evaluate:
∫∫exp((y-x)/(y+x))dxdy for D
by making the substitutions u=y-x and v=y+x
Homework EquationsThe Attempt at a Solution
So first I found an equation for y and x respectively:
y=(u+v)/2 and x=(v-u)/2
Then...
Homework Statement
If I have the two curves
##\phi (t) = ( \cos t , \sin t ) ## with ## t \in [0, 2\pi]##
##\psi(s) = ( \sin 2s , \cos 2s ) ## with ## s \in [\frac{\pi}{4} , \frac{5 \pi}{4} ] ##
My textbook says that they are equivalent because ##\psi(s) = \phi \circ g^{-1}(s) ## where ##...
I am studying Analysis on Manifolds by Munkres. He introduces improper/extended integrals over open set the following way: Let A be an open set in R^n; let f : A -> R be a continuous function. If f is non-negative on A, we define the (extended) integral of f over A, as the supremum of all the...
1. The problem statement, all variables, and given/known data
Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##.
Homework Equations
Chain rule $$\frac{dz}{dx} = \frac{dz}{dy}...
I am not sure how I should set my u and v expressions into the u-v plane for this question.
How should I look at the expression to set u and v expressions?
I have obtained as such using changing of variables/ transformations:
Let u= y/x and v= xy after manipulating the equation of the curves as such xy = 5, xy = 2, y/x = 1 and y/x = 4.
Then u = 1, u = 4, v = 5 and v = 2 => obtain a square on the u-v plane.
Change the integrand to:
Outer...
Hey! :o
Let $D$ be the space in the first quadrant of the $xy$-plane that is defined by the inequality $x^{\frac{3}{2}}+y^{\frac{3}{2}} \leq \alpha^{\frac{3}{2}}$ with $\alpha>0$. I want to transform $\iint_D f(x,y) dx dy$ to an integral on the triangle $E$ of the $uv$-plane that is defined by...
Evaluate ∫∫∫ over E, where E is the solid enclosed by the ellipsoid
x^2/a^2 + y^2/b^2 + z^2/c^2 = 1.
Use the transformation x = au,
y = bv, z = cw.
I decided to replace x with au, y with bv and z with cw in the ellipsoid.
After simplifying, I got
u^2 + v^2 + w^2 = 1.
What is the next step...
Let S S = double integrals
S S x^2 dA; where R is the region bounded by the ellipse
9x^2 + 4y^2 = 36.
The given transformation is x = 2u, y = 3v
I decided to change the given ellipse to a circle centered at the origin.
9x^2 + 4y^2 = 36
I divided across by 36.
x^2/4 + y^2/9 = 1
I replaced...
Homework Statement
Hi,
I am looking at this question:
With this (part of ) solution:
Homework EquationsThe Attempt at a Solution
I follow up to the last line-
I do not understand here how we have simply taken the ##1/t^{\alpha m + \alpha}## outside of the derivative...
Homework Statement
I am trying to minimize the function ##f(a) = (1+4a^2)^3 \left( \frac{1}{4a^2} \right)^2##. Here we are given that ##a>0##
Homework Equations
Definition of a minimum of a function
The Attempt at a Solution
Now the derivative here will be ugly and equating it to zero and...
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
I solved part a. I got an answer of 140. For part b, however, I am stuck. I came up with a set of points for D in the xy plane [(0,3)(0,6)(4,5)(4,8)] giving me a rhombus. How do i integrate this? I tried to split up the...
Homework Statement
Determine the spherical harmonics and the eigenvalues of \vec{\hat{L}}^2 by solving the eigenvalue equation \vec{\hat{L}}^2 |\lambda, m \rangle in position space,
[\frac{1}{sin \theta} \frac{\partial}{\partial \theta} ( sin \theta \frac{\partial}{\partial \theta} ) +...
Hi All,
$$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$
I can perform the integration of the integral above easily by changing the variable
$$u=x_2+x_1\\
v=x_2-x_1$$
Of course first computing the Jacobian, and integrating over ##u## and ##v##
I am wondering how you perform the change of...
Q: Suppose ##u(x,t)## satisfies the heat equation for ##0<x<a## with the usual initial condition ##u(x,0)=f(x)##, and the temperature given to be a non-zero constant C on the surfaces ##x=0## and ##x=a##.
We have BCs ##u(0,t) = u(a,t) = C.## Our standard method for finding u doesn't work here...
We start with:
d2y/dx2
And we want to consider x as function of y instead of y as function of x.
I understand this equality:
dy/dx = 1/ (dx/dy)
But for the second order this equality is provided:
d2y/dx2 =- d2x/dy2 / (dx/dy)3
Does anybody understand where is it coming from? The cubic...
Homework Statement
Hi guys, I'm struggling to figure out how the solution in the picture that I posted was able to get rid of their mg factors and then come up with a factor of k for x_1 in their eigenvalue equation. You can see that in the second equation of motion there is no k*x_1 but it...
Homework Statement
Transform the equation:
x2 * d2y/dx2 + 2 * x * dy/dx + (a2/x2)*y = 0
Using:
x=1/t
Homework Equations
The differential of a function of several variables, and the common rules of differentiation.
https://en.wikipedia.org/wiki/Derivative
The Attempt at a Solution
As...
With the change of variables-method for a many-to-one transformation function Y = t(X),
what's the logic behind summing the different densities for the roots of x = t^-1(y)?
Probabilities should be ok to add, but densities?
Also, is there no way to extend this method for many-to-many...
I get two different answers, ##a^2## and ##0.5a^2##, by using two different methods. Which is the correct answer?
The family of curve for ##y^2=4u(u-x)## is given by the blue curves, and that for ##y^2=4v(v+x)## is given by the red curves.
Method 1:
Evaluate the integral ##I## directly in...
I am trying to prove that the above is true when performing the change of variable shown. Here is my attempt:
What I am not quite understanding is why they choose to isolate the partial derivative of ##z## on the right side (as opposed to the left) that I have in my last line. This ultimately...