# 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.

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.

Then, assuming that one is interested only in real solutions, the solutions of the original equation are

x
=
(
1

)

1

/

3

=
1

and

x
=
(
8

)

1

/

3

=
2.

View More On Wikipedia.org

3. ### I Quick Change of Variables Question

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)...

13. ### Solving the wave equation with change of variables approach

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...
14. ### I Change of Variables in Thermodynamics

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...
15. ### Solving an inequality for a change of variables

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...
16. ### Change of Variables: Integrals w/Polar Coordinates

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...
17. ### Probability: Multivariate distribution change of variables

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 ##...
18. ### I Understanding the change of variables for PDEs

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...
19. ### I Change of variables in the Density of States function

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...
20. ### Double Integral via Appropriate Change of Variables

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...
21. ### Change of variables in a propagator

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...
22. ### Change of variables in a simple integral

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...
23. ### I Change of variables for this derivative in a heat transfer equation

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...
24. ### I PDEs: Diffusion Equation Change of Variables

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...
25. ### Solving Variable Change: Difficulties Understanding ##V(x)## & ##y##

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...
26. ### I Multiple integral Jacobian confusion

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'}...
27. ### Change of variables on autonomous systems solutions

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...
28. ### Diff.equation transformation by change of variables

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...
29. ### I Change of variables; why do we take the absolute value?

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...
30. ### Vector Calculus: Change of Variables problem

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...
31. ### Curve and admissible change of variable

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 ##...
32. ### I Munkres-Analysis on Manifolds: Extended Integrals

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...

44. ### Change of variables in Heat Equation (and Fourier Series)

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...
45. ### I Chain rule and change of variables again

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...
46. ### Change of variables and gravity constants

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...
47. ### Change of variables in a differential equation

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...
48. ### I Change of variables many-to-many transformation

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...
49. ### I Change of variables in double integral

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...
50. ### I Change of variable - partial derivative

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...