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 Apr18 comment Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. Is there an error in this solution? Not sure why it has a downvote as it looks correct to me. Apr12 comment Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$. Thanks for clearing up my understanding. Apr12 comment Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$. Don't we also have that $y$ is a function of $x$ and vice versa when implicitly differentiating? (i.e. $y(x)$ so requires the product rule when differentiating $yf'(a)$?) Apr12 comment Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? Thanks for this. What is the best way to solve the system of equations by hand? At first glance, it seems quite messy. Apr11 comment What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$? This is a good answer also. Although, I see we want to set $d/dx (d^2) =0$ and solve for $t$. What is the best way to do this by hand? It again seems cumbersome. 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 $g(0)=0$? 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 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 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 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 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 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 $D= \{ x \in \mathbb{R}^3 : xyz \neq 0\}$? Mar25 comment Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why? "... the set maps $A \mapsto \int_A f(x) dx$ are actually measures, so you can use...". Can you elaborate a little on this part? What do you mean by continuity of measure and why can we use it to recover the result of the improper Riemann integral? Perhaps it would be good to know what theorems are at play here. Mar25 comment Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why? Do you mean $\int_{2^{-n-1}}^{2^{-n}}$ and $\mu([2^{-n-1},2^{-n}))$ since $2^{-n-1} < 2^{-n}$? Also, can we use a similar process with a simple function to show that $1/\sqrt{x}$ is Lebesgue integrable? What would the simple function be? Mar25 comment Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$ how can one show that the integral of the simple function approaches infinity? Mar21 comment If $f,g: R^n \to R^3$, what is the derivative of the cross product $(f \times g)(\vec{a})$ where $\vec{a} \in R^n$? Yes, corrected. For $n > 1$, I'm wondering if there is an analogous statement for functions of several variables. Mar17 comment What is the limit of $f(a,b) = \frac{a^\beta}{a^2 + b^2}$ as $(a,b) \to (0,0)$? @EclipseSun Apologies. I've explained in the comments to the question. I will proceed with more consideration next time around. Mar17 comment What is the limit of $f(a,b) = \frac{a^\beta}{a^2 + b^2}$ as $(a,b) \to (0,0)$? To explain, I've been creating different functions with $\beta$ to understand certain outcomes. As it happened, the answer to my original formulation didn't help my understanding. In retrospect, I see it's caused a mess and I should have asked a second question rather than done a modification (which I made because I didn't want to flood the site with similar questions).