Recent content by kalvin

Find the integral of the function $f(x,y) = \frac{1}{(x^2 + y^2 + 1)^{\frac{3}{2}}} $ over the closed ball $\overline{B(a, 2)}$(i.e disk with radius 2 centered at point a). Letting $a \rightarrow \infty$, show that: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}f(x,y) dydx = 2\pi$

Hi, I was just wondering if it was typo for z. Should z = (a, b) ? and not (a, c)? If it is not a typo do you think you could explain? Thank you

Let $U \subset \Bbb{R}^3$ be open and let $f : U → \Bbb{R}$ be a $C^1$ function. Let$ (a, b, c) \in U$ and suppose that$ f(a, b, c) = 0$ and $D_3f(a, b, c) \ne 0.$ Show there is an open ball$ V \subset \Bbb{R}^2$ containing $(a, b)$ and a $C^1$ function $\phi : V → \Bbb{R}$ such that $\phi(a, b)...

Let $K \subset \mathbb{R^n}$ be compact and let $f: K \rightarrow \mathbb{R}$ be continuous. Suppose that $f(x) > 0$ $\forall x \in S.$ Prove there is a $c > 0$ such that $f(x) \geq c$ $\forall x \in K$ My Sol: I said that by the extreme value theorem $\exists a,b \in K $ such that $f(a)...