This concern all problem requesting techniques and tricks about changes of variables in both computation of limits and integrals
Questions tagged [change-of-variable]
1125 questions
3
votes
1 answer
Is there always a currency that goes up compared to all others?
Suppose you go through every country's currencies and look at the change WRT all other currencies that day. Assuming there is a change in value that day, does it stand to reason that there will always be a certain currency that goes up WRT all…
Derek Seabrooke
- 131
3
votes
2 answers
Understanding change of variables in double integrals
In this libre text chapter, in example 15.7.1B, the author illustrates the change of variables using the following example:
The statement near it is written:
For the vertical length $A:u=0,0≤v≤1$ transforms to $x=−v^2,y=0$ so this is the…
tryst with freedom
- 11,538
1
vote
1 answer
Confused about types of variable substitutions
I am confused about (dummy?) variable substitutions in comparison with change of variable type of substitutions.
On the one hand, I can write $f(x) = \dfrac{1}{1+x}$ or $f(u) = \dfrac{1}{1+u}$ and the functions produce same result for equal values…
Kavka
- 341
1
vote
5 answers
Let $a,$ $b,$ $c$ be real numbers such that $abc = 1$ and $\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} = 0.$ Find $a + b + c.$
Let $a,$ $b,$ $c$ be real numbers such that $abc = 1$ and $\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} = 0.$ Find $a + b + c.$
I don't really know how to approach this. I was thinking doing something like squaring or cubing $\sqrt[3]{a} + \sqrt[3]{b} +…
Mike Smith
- 1,122
- 6
- 26
0
votes
0 answers
Differentiation of the Norm in $\mathbb{R}^n$
I'm reading Evan's Partial Differential Equations. In the proof of the Theorem 1(iii) (Initial Value Problem of the Heat Equation on Page 48), the last step states…
IntegralLover
- 382
0
votes
1 answer
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 $x\mapsto y=\varphi(x)$. If I know that the Jacobian of my change of…
Marrie
- 85
0
votes
1 answer
Bijective change of variable
Suppose we have a function $f: \theta\rightarrow f(\theta)$ whose domain is $[0,180°]$. The course I follow, says that we can use the change of variable $x=cos(\theta)$ in the function $f$ (because $cos$ is bijective from $[0,180°]$ to $[-1,1]$).
I…
niobium
- 1,177
0
votes
0 answers
How to change the variables in this integral?
I want to calculate the following integral
$$
\int_0^{\infty}\int_0^{\infty}\frac{f(r_1,r_2)}{\lvert r_1-r_2\rvert}\; r_1^2 r_2^2\; dr_1 dr_2 \tag{1}
$$
also I intend to use the following change variables
$$
r=\lvert r_1-r_2\rvert…
Wisdom
- 395
0
votes
1 answer
What role does dummy variable play in this example?
It's an IVP in Separation Equations :
$$\frac {dy}{dx}=e^{-x^2}, y(3)=5$$
Here is the solution on text:
$$\int^x_3\frac {dy}{dt}dt=\int_3^xe^{-t^2}dt$$
It comes out
$$y(x)=y(3)+\int_3^xe^{-t^2}dt$$
And here is my answer with…
0
votes
1 answer
Leibniz notation for change of variables in probability
Here is a really basic problem. Let $R$ be the radius of a sphere, and draw this uniformly on $[1,10]$ so that $R$ has density $$f_R(x)=\frac{1_{1
mostsquares
- 940
0
votes
1 answer
Change of variables in double integrals intuitively
So I have been trying to understand the intuitive proof of change of variables in double integrals.
Equations $$ x=x(u, v) \ \mathrm{and} \ y=y(u,v) $$
define a mapping from uv-Cartesian plane to points in xy-plane. The mapping is one-to-one.
The…
mathslover
- 1,482
0
votes
1 answer
How to clear $T$ from the formula $vf = vp ( 1 + t \times i )$?
Given the formula $vf = vp ( 1 + t \times i )$, I would like to clear $t$.
I would like to know what is the steps to do this. I have issues with formulas and don't understand. Please help me.
Professor Zoom
- 113
- 5
0
votes
1 answer
How do you prove loss and gain?
Why is gain equal to final value minus initial value, loss equal to initial value minus final value? How do you prove them mathematically?
0
votes
1 answer
Change of integration variable
I have the following relation for a function $\phi_A(\textbf{r})$ and it is known that $\rho_B(\lambda \textbf{r}) = \rho_A(\textbf{r})$
$\phi_A(\textbf{r}) = \int{d^3r' \frac{\rho_A(\textbf{r}')} {|\textbf{r - r}'|}} = \int{d^3r'…
Paddy
- 457
0
votes
1 answer
Change of variable in distribution
I am a physics student. I am studying the Maxwell-Boltzmann velocity distribution. Initially I began working with the distribution expressed in terms of momentum $p$, lets coll it $f_p(p)$, which is such that
$$ n = f_p(p) \ dp $$
is the number of…
user171780
- 119