Question

@ganeshie8

Answer 1

How can I show the shortest distance between two points in R3 space is a straight line?

Answer 2

I know it has to do with \[dS = \sqrt{dx^2+dy^2+dz^2}\]
and maybe Euler's equation

Answer 3

I'm thinking we let it depend on the parameter so maybe then we can define our function as \[f(t) = \sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2}\]

Answer 4

So Euler's equation which is \[\frac{ \partial f }{ \partial y } - \frac{ d }{ dx } \frac{ \partial f }{ \partial y' } = 0\] right, so if we take the partial of f respect to x,y, and z we'll get 0. I think this is working out nicely, I'm going to see what I get as a final result

Answer 5

Are you kindof trying to find a path that minimizes the arc length integral ?

Answer 6

Yup!

Answer 7

This looks interesting

Answer 8

If possible can you put the complete work when u finish

Answer 9

Yeah sure, give me a bit, it might take me a while haha.

Answer 10

I'm trying to do this problem, as I've been learning calculus variation and what not, it's very fascinating stuff.

Answer 11

You gona solve that differential eqn next is it

Answer 12

Find a curve \(r(t)\) such that below arclength integral is minimum :

\[ L= \int_a^b \|r'(t)\|dt\]

Answer 13

why not start with showing the 2-D case

Answer 14

yeah im working on 2D case on my notes

Answer 15

kai made this tutorial a while back about euler-lagrange method one of its uses is to minimize integrals like this

Answer 16

http://openstudy.com/study#/updates/53b2884ae4b09f140ca3f90a

Answer 17

I did 2d

Answer 18

\[dS = \sqrt{dx^2+dy^2}\]
\[f = \sqrt{1+\left( \frac{ dy }{ dx }\right)^2}\] then you can use Euler's equation, and pretty simple from there, and you will get an equation of a straight line, I'll let you finish it on your own, or if you want the full work just ask. I'm sort of doing the same thing with space

Answer 19

\[\int\limits_{x_1}^{x_2} \sqrt{1+ \left( \frac{ dy }{ dx } \right)^2dx}\] we let S be the minimum so we can define the function above, i think that's the hardest part about the problem really

Answer 20

S ^

Answer 21

Neat huh? :P

Answer 22

Oh I just looked at the post dan made, kai actually did it for 2d, pretty much the same exact thing.

Answer 23

http://puu.sh/gvtSk/18243df84c.png

Answer 24

Since we're trying to minimize the value of definite integral of \(f(x,y,y')\) between two points,

\[df = f_x dx + f_y dy+f_{y'} dy' \implies \dfrac{df}{dx} = f_x + f_y y' + f_{y'} y'' \tag{1}\]

eliminating \(y''\) we get
\[ \dfrac{d}{dx}\left(f-y'\dfrac{\partial f}{ \partial y' } \right) - \dfrac{\partial f}{\partial x} = 0\]

Answer 25

Ok so for space, \[\frac{ d }{ dt } \frac{ \partial f }{ \partial x' } \implies \frac{ x' }{ \sqrt{x'^2+y'^2+z'^2} } = C\] we can do this with y and z as well, so we get all the constants.
\[\frac{ x' }{ C } = \frac{ y' }{ C_2 }\] doing the same thing with z, we can integrate and get a equation for a straight line in R3 I'm sure

Answer 26

nicee

Answer 27

Yeah I think it works out pretty well :P

Answer 28

Quick question, how do I add the dots for derivatives instead?

Answer 29

I think dots are not allowed in math group, you should use primes/subscripts for derivatives/partials :P

Answer 30

\[\huge \dot{y}\]

Answer 31

Yeah well we use dots in physics :P, this is more so a physics question I think?

Answer 32

Oh ok, haha thanks!

Answer 33

but you're in math group hmm

Answer 34

Haha xD

Answer 35

Ok prime is fine

Answer 36

I got more questions, tougher than this...if you want to have a try at them, I'll be trying them later tomorrow

Answer 37

Or I'll just post the questions I get stuck on :P

Answer 38

sure tag me in all questions, i remember doing few of these in my engineering math.. but really don't remember much.. il check out paul site and read some more in the evening

Answer 39

Ok cool :P, I think it's about time I give you tough questions rather than the easy calc ones hehe.

Answer 40

You should try to derive Euler's formula, it will be very useful, especially for these types of problems

Answer 41

i derived it already in my reply above using total derivative

do you have a pdf/something thats really short to read ?

Answer 42

http://gyazo.com/0097701d005ec32fdb4761174e36851c

this is equivalent to euler eqution

Answer 43

I see, interesting and not really, I've been just reading my book, but I'm sure there are many calculus variation pdfs out there