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
define L[u] = a \frac{\partial^2u}{\partial t^2} + B \frac{\partial^2 u}{\partial x \partial t} + C \frac{\partial^2u}{\partial x^2} = 0
show that if L[u] is hyperbolic then and A is not zero the transofmartion to moving coordinates
x' = x - \frac{B}{2A} t
t' = t
tkaes L into a...
Hmm, I can't seem to get this double integral transformation:
int(limits of integration are 0 to 3) int (limits of int are 0 to x) of (dy dx)/(x^2 + y^2)^(1/2)
and i need to switch it to polar coordinates and then evaluate the polar double integral.
i sketched the region over which i am...
Hi, I'm having trouble understanding the solution to a question from my book. I think it's got something to do with the chain rule. The problem is to prove the change of variables formula for a double integral for the case f(x,y) = 1 using Green's theorem.
\int\limits_{}^{} {\int\limits_R^{}...
Hi, I'm having trouble evaluating the following integral.
\int\limits_{}^{} {\int\limits_R^{} {\cos \left( {\frac{{y - x}}{{y + x}}} \right)} } dA
Where R is the trapezoidal region with vertices (1,0), (2,0), (0,2) and (0,1).
I a drew a diagram and found that R is the region bounded...
let f be continuous on [0,1] and R be a triangular region with vertices (0,0), (1,0) and (0,1). Show:
the double integral over the region R of f(x+y)dxdy = the integral from 0 to 1 over u f(u)du
I recognize it is a change of variables problem but I'll be damned if I can create a set of...
I asked this in another thread, but I think this forum might be a better place for it (not trying to spam the same question). When deriving the formula for relativistic kinetic energy, we start with
KE = \int_{0}^{s} \frac{d(mv)}{dt} ds = \int_{0}^{mv} v d(mv)
So I figure that since v =...
let be the integral:
\int_1^{\infty}\int_{-\infty}^{\infty}f(x,y)dydx
i make the change of variable xy=u y=v whose Jacobian is 1/v but then what would be the new limits?...
Hi,
I'm not sure how to do this question. Any help would be great.
Let B be the region in the first quadrant of R^2 bounded by xy=1, xy=3, x^2-y^2=1, x^2-y^2=4. Find \int_B(x^2+y^2) using the substitution u=x^2-y^2, v=xy. . Use the Inverse Function theorem rather than solving for x...
Hi. I have a problem with a question. Basically, I have an integral that goes from x=0 to x=1, and I'm supposed to make a change of variables like this:
Let x = 1 - y^2.
The problem I'm having is trying to find the limits of integration after the change of variables. Since y = +/-...
I'm trying to evaluate the double integral
\int \int \sqrt{x^2 + y^2} \, dA
over the region R = [0,1] x [0,1]
using change of variables.
Well, after fooling around, I've got an answer. I set u = x^2, v =y^2, and then calculated the jacobian of T which was 1. The image transformation...
I'm trying to evaluate the double integral
\int \int \sqrt{x^2 + y^2} \, dA over the region R = [0,1] x [0,1]
using change of variables
Now I know polar coordinates would be the most efficient way, and thus I could say r= \sqrt{x^2 + y^2} . Is this legal to use polar coordinates...
Can anyone give me any hints as to find a suitable change of variables for this integral.
infinity
/
|dt/(a^2+t^2)^3/2 =
|
/ -infinity
=2/a^2 * integral below...
Let R be the region bounded by the graphs of x+y=1, x+y=2, 2x-3y=2, and 2x-3y+5. Use the change of variables:
x=1/5(3u+v)
y=1/5(2u-v)
to evaluate the integral:
\iint(2x-3y)\,dA
I found the jachobian to be -1/5
and the limits of integration to be
1<=u<=2
2<=v<=5
so i set up...
Ok, i have a problem with this double integral. I am having a hard time finding the limits. The question is
Evaluate
\iint \frac{dx\,dy}{\sqrt{1+x+2y}}\
D = [0,1] x [0,1], by setting T(u,v) = (u, v/2) and evaluating the integral over D*, where T(D*)=D
Can some one help me find the...
Does anyone know of any sources that explain change of variables for double integrals. Actually, I get the change of variables thing, but a few of our problems don't give us the transforms. I don't understand how to create these myself.
Here is an example:
Math Problem
So far, I found...
Wacky change of variables for Multi integration!
Arghh I am having diffiiculty with these problems.
I am having difficulty mastering the LaTeX form--- (things like how to make a double integral etc) so
if you look at this site...
im working on these, and I am supposed to find the image of a set under a given transformation. can someone please explain to me a good way of doing this?