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 Apr8 accepted Prove that any bounded open set has an arbitrarily close closed subset Apr8 accepted Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why? Apr8 accepted Show that a system of equations can be solved in terms of $x,y,z$ (Rudin, ex 9.19) Apr8 comment Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$. It cannot be done with this assumption that $g(0)=0$? Apr8 asked Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$. Apr5 comment Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$ This is indeed the proof I've seen. Thanks for the reference. Apr5 accepted Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$ Apr5 asked Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$ Apr5 revised Show that a system of equations can be solved in terms of $x,y,z$ (Rudin, ex 9.19) added 3 characters in body Apr5 comment Show that a system of equations can be solved in terms of $x,y,z$ (Rudin, ex 9.19) Thanks. I understand this part of the question. What I'm not sure about is why x=(-9-z)/z if u=3. Is that the answer you get as well? Apr5 asked Show that a system of equations can be solved in terms of $x,y,z$ (Rudin, ex 9.19) Apr5 comment What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$? Question about your solution: Say I want to make restrictions on (x,y,z) such that $f(x,y,z)$ is injective and surjective throughout its entire domain (not just locally). Is it enough to restrict each of $x,y,z$ to either it's positive or negative part such that there are 8 possible domains where $f$ is injective and surjective throughout? Apr4 comment What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$? sure thats fine Apr4 accepted What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$? Apr3 comment What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$? Thanks. I will digest this but from what I can see, your approach is to find the inverse Jacobian first because it seems easier. So, there are six different values for $g$ -- I assume this is because of the squares of $x,y,z$ in $f$. What method did you use to find $g$? Substitution? Apr3 comment What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$? For an invertible matrix? Yes. Or are you saying there's a closed for for the inverse of the function? I think it's finding an explicit form for the inverse of the function that is my real problem. Apr3 asked What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$? Apr3 comment What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$? Right. I meant if we restrict the domain to $D$ in my previous comment AND we have that $\alpha\gamma\varphi + \beta\delta\epsilon \neq 0$, then $f$ is injective and surjective on $D$. Apr3 accepted What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$? Apr3 comment What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$? Thanks. Does this mean that $f$ injective and surjective if we restrict the domain to $D= \{ x \in \mathbb{R}^3 : xyz \neq 0\}$?