1. line integrals

may i know how to solve this question? i tried it , as seen in the picture, but got stuck

2. Originally Posted by alexandrabel90

may i know how to solve this question? i tried it , as seen in the picture, but got stuck
$\frac{d}{dt}\frac{1}{\sqrt{x^2+y^2+z^2}}=-\frac{1}{2\cdot (x^2+y^2+z^2)^{3/2}}\cdot (2x \frac{dx}{dt}+2y\frac{dy}{dt}+2z\frac{dz}{dt})=\ld ots$

So $\frac{1}{\sqrt{x^2+y^2+z^2}}$ is an anti-derivative of the integrand in question...

3. how were you able to deduce from this integration that the equation that you wrote is the anti-derivative of the integrand in question?

thanks!

4. Originally Posted by alexandrabel90
how were you able to deduce from this integration that the equation that you wrote is the anti-derivative of the integrand in question?

thanks!
To be quite honest with you it was primarily because this is a famous example of a "central force" being conservative (gravitation), and I knew that in the one-dimensional case, it is quite obvious that this amounts to $\int -\frac{1}{r^3}\cdot r\, \frac{dr}{dt}\, dt=-\int\frac{1}{r^2}\,dr = \frac{1}{r}+C$.
The three-dimensional case is a little less obvious than this, but it also amounts to a simple substitution of $r := \sqrt{x^2+y^2+z^2}$, checking that $dt$ can be gotten rid of and then the integral really does turn out to have a value that is dependent on $r$ only, and therefore only on the location of the end-points of the path.

Moral of the story: Nothing beats memory when it comes to problem-solving . I am told that all experts mainly rely on memory (aka. experience), and use reasoning only when they absolutely have to.