All Questions
123
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A 4d integral, Exercise I-8 p.68-69 from "Mathematics for the physical sciences" (Dover), Laurent Schwartz
The first part is about two versions of a formula by Feynman for the inverse of a product, one of which being
$$ \frac{1}{a_1\, a_2 \, \cdots\, a_n} = \int_0^1 \int_0^{1-x_1} \cdots \int_0^{1-x_1 -x_2 ...
- 3,679
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Expressing integral based on the change of variables in double integral
Let B be the region in the first quadrant bounded by the curves $xy=1$,$xy=3$,$x^2−y^2=1$, and $x^2−y^2=4.$
Evaluate $\iint_{B} (x^2 + y^2 ) dx dy$ using the change of
variables $u=x^2−y^2$,$v=xy.$
...
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The last part of Step 5 in the proof of change of variables theorem (Lemma 19.1) in "Analysis on Manifolds" by James R. Munkres.
I am reading "Analysis on Manifolds" by James R. Munkres.
I cannot understand the last part of Step 5 in the proof of change of variables theorem (Lemma 19.1)
https://archive.org/details/...
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$h(D\times\{x^n\})\subset\mathbb{R}^n$ and the author wrote $\int_{h(D\times\{x^n\})} 1 dx^1\cdots dx^{n-1}$. Is this ok? Calculus on Manifolds Spivak
I am reading "Calculus on Manifolds" by Michael Spivak.
In the proof of Theorem 3-13 (Change of variable), the author wrote as follows:
Let $W\subset U$ be a rectangle of the form $D\times [...
- 1,138
1
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0
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Why did the author calculate $D_n(g^n\circ h^{-1})(h(a))$ to conclude $k'(h(a))=I$? ("Calculus on Manifolds" by Michael Spivak.)
I am reading "Calculus on Manifolds" by Michael Spivak.
In the proof of Theorem 3-13 (Change of variable), the author wrote as follows:
Since $$(g^n\circ h^{-1})'(h(a))=(g^n)'(a)\cdot [h'(a)...
- 1,138
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I wonder why we need (2) to prove the change of variable theorem for the case $n=1$. ("Calculus on Manifolds" by Michael Spivak.)
I am reading "Calculus on Manifolds" by Michael Spivak.
In the proof of Theorem 3-13 (Change of variable), the author wrote as follows:
We are now prepared to give the proof, which proceeds ...
- 1,138
1
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0
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How to show $\int_{h(g(A))} f=\int_{g(A)} (f\circ h)|\det h'|$ holds when $\int_{h(B)} f=\int_{B} (f\circ h)|\det h'|$ Calculus on Manifolds Spivak.
3-13 Theorem. Let $A\subset\mathbb{R}^n$ be an open set and $g:A\to\mathbb{R}^n$ a 1-1, continuously differentiable function such that $\det g'(x)\neq 0$ for all $x\in A$. If $f: g(A)\to\mathbb{R}$ is ...
- 1,138
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Understanding the Meaning of $y \geq x$ in Cylindrical Coordinates during a Variable Transformation.
I am seeking clarity / intuition on the meaning of the condition $y \geq x$ when performing a change of variables from Cartesian to Cylindrical Coordinates for volume integration.
In the context of ...
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88
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Change of variables on integral of multivariate gaussian
I have a question on the change of variables for multivariate Gaussian integrals.
Suppose that we have two independent random variables $X_1$ and $X_2$ where $X_i \sim N(0, \sigma_i^2)$ for $i=1,2$. ...
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Stuck at understanding how to properly use the change of variables theorem when localizing the smooth bump function to a given open ball
This is question is spin-off of my prior question: Trouble with change of variables when constructing a smooth bump function localized on a given open ball, but this one is more suited to ...
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Change of variables to evaluate a double integral - limits of integration [closed]
Suppose I want to compute
$$ I\equiv \int_R (x+y) \exp(x-y) \, dx\, dy$$
where the integration region $R$ is the rectangle delimited by $(a,b),\,(c,b),\,(c,d),\,(a,d)$, so namely
$$ I\equiv \int_a^c ...
- 171
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86
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Finding the area bounded by some curves using change of variables and double integrals
find the area of the first-quadrant region bounded by the curves $y = x^3,y=2x^3,x=y^3,x=4y^3$ be the curves that bound the area:
now we can use the substitution:
$$y =ux^3$$
$$x =vy^3$$
in order to ...
- 1,043
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40
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Help with a variable change
I'm integrating a function of the variables $x_i,y_i,z_i$ for $i=1,2,3$ and I define new variables given by
$$u_1=x_1+y_3 z_3,$$
$$u_2=x_2+y_1 z_1,$$
$$u_3=x_3+y_2 z_2.$$
NOTE: I don't actually want ...
6
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2
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Find the volume of the solid $Q$ cut from the sphere $x^2 + y^2 + z^2 ≤ 4$ by the cylinder $r = 2 \sin θ$ .
I set up the integral as $$\int_0^{\pi}\int_0^{2\sin\theta}\int_{-\sqrt{4-r^2}}^{\sqrt{4-r^2}} r dzdrd\theta$$ and got $\frac{16\pi}3$, but the answer is $\frac{16(3\pi-4)}9 \approx 9.64$, which is ...
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Infinite Integrals and "Moments" of a Random Vector
I'm trying to understand the concept of "moments" of random vector, and what this means for (potentially) infinite integrals. Here is an example
Suppose we have a random vector $\alpha$ ...
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