problem

Posted July 26th, 2009 by Structure

List of derivatives

Posted February 9th, 2009 by Structure

We have

$$f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x}$$

For many elementary functions this type of limit is very easy to be found. We need some algebra but sometimes you have to use some remarkable limits.

$$(x^n)'=nx^{n-1}$$

as

$$\lim_{y\to x}\frac{y^n-x^n}{y-x}=\lim_{y\to x}y^{n-1}+y^{n-2}x+...+yx^{n-2}+x^{n-1}=nx^{n-1}$$
$$(lnx)'=\frac{1}{x}$$
$$(\sqrt x)'=\frac{1}{2\sqrt x};\lim_{y\to x}\frac{\sqrt y-\sqrt x}{y-x}=\lim_{y\to x}\frac{1}{\sqrt y+\sqrt x}=\frac{1}{2\sqrt x}$$

Surface integral in spherical coordinates

Posted January 29th, 2009 by Structure

Let us consider a surface integral

$$\int_{\Sigma}F(x,y,z)d\sigma$$

where $ \Sigma $ is a surface which have a parameterization described in terms of angles $ \theta $ and $ \phi $ in spherical coordinates.
Let
$ x=r \cos \theta sin \phi $
$ y=r \sin \theta \sin \phi $
$ z = r \cos \phi $
and let
$ \chi(r,\phi,\theta)=(r \cos \theta sin \phi, r \sin \theta \sin \phi,r \cos \phi) $
We are interested in a formula for evaluating a surface integral where r is a function of angular variables
$ r=\rho(\theta,\phi) $

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