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3h
accepted Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.
Apr
12
accepted Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$.
Apr
12
accepted What is the shortest/longest distance from $9x^2 + 4y^2 = 36$ to $(5,5)$?
Apr
12
accepted Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?
Apr
11
accepted How to solve $3y\cos(\theta) - 2x\sin(\theta) = 5 \sin(\theta)\cos(\theta)$?
Apr
8
accepted Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$.
Apr
8
accepted Prove $f(x,y) = \frac{x^2+y^2}{x+y}$ is not continuous at $(0,0)$.
Apr
8
accepted Prove that if $E$ is measurable then $\forall \epsilon > 0$ $\exists F \subset E$ closed such that $m(E \setminus F) < \epsilon$.
Apr
8
accepted Prove that any bounded open set has an arbitrarily close closed subset
Apr
8
accepted Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why?
Apr
8
accepted Show that a system of equations can be solved in terms of $x,y,z$ (Rudin, ex 9.19)
Apr
5
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$\quad\Big]$
Apr
4
accepted What are the inverse function and inverse Jacobian of $f(x,y,z) = \left(\frac{ix^2 + hy^2}{2}, \frac{jy^2+kz^2}{2}, \frac{mx^2 + nz^2}{2} \right)$?
Apr
3
accepted What is needed to apply the inverse function theorem to $f(x,y,z) = \left(\frac{ax^2 + by^2}{2}, \frac{cy^2+dz^2}{2}, \frac{ex^2 + fz^2}{2} \right)$?
Mar
29
accepted Why are there different methods of constructing the outer measure leading up to Lebesgue measure (open covering or closed covering?)?
Mar
25
accepted If $f,g: R^n \to R^3$, what is the derivative of the cross product $(f \times g)(\vec{a})$ where $\vec{a} \in R^n$?
Mar
18
accepted Does $\int_0^\infty f(x) dx = \lim_{n \to \infty} \int_0^n f(x) dx$ for $f \geq 0$ or $f$ not positive?
Mar
17
accepted What is the limit of $f(a,b) = \frac{a^\beta}{a^2 + b^2}$ as $(a,b) \to (0,0)$?
Mar
16
accepted Show that $-\frac{2yx^3}{(x^2+y^2)^2}$ is bounded.
Feb
19
accepted When writing the integral sign $\int$, how does one know what integral is being discussed?