4 votes

Integrating $e^{-(x^2+y^2+z^2)/a^2}$ over $\mathbb{R}^3$

This integral factors as $$\left(\int_\mathbb{R}e^{-x^2/a^2}dx\right)^3$$ The integral in the parentheses is well-known, and is called a Gaussian integral. Here it is explained many methods of how to ...
Joshua Tilley's user avatar
3 votes

About the continuity of the partial derivative

If you really don't like trigonometric functions, we can consider the rectangles instead; however, this is not really any different than Ted Shifrin's answer. Consider the family of rectangles $$ \max(...
robjohn's user avatar
  • 345k
1 vote

Evaluate integral of $\langle xz^2,x^2y-z^3,2xy+y^2z\rangle$ with divergence theorem

The integral in your post is over a closed volume bounded by half a sphere of radius $a$. By divergence theorem, $$\begin{aligned} \oint_{S}\langle xz^2,x^2y-z^3,2xy+y^2z\rangle\cdot\mathrm d\mathbf S&...
Joan S. Guillamet F.'s user avatar
2 votes

About the continuity of the partial derivative

In problems like this I find it useful to use a homogenized form of polar coordinates: Let $x^4=r\cos\theta$ and $y^2=r\sin\theta$, with $0\le\theta\le\pi/2$. Then $$\left|\frac{x^3y^3(y^4-x^8)}{(y^4+...
Ted Shifrin's user avatar
1 vote

Integrating $e^{-(x^2+y^2+z^2)/a^2}$ over $\mathbb{R}^3$

If you really want to use spherical coodinates, then you can proceed as follows. $$\begin{align} I&=\int_{\mathbb{R^3}}e^{-(x^2+y^2+z^2)/a^2}\,dx\,dy\,dz=\int_0^{2\pi}\int_0^\pi \int_0^\infty e^{-(...
Mark Viola's user avatar
  • 179k

Only top scored, non community-wiki answers of a minimum length are eligible