## xapproachesinfinity one year ago guys! here is something. my friend asked me a question if we increased the length of an object by x how much its area has increased. so my answer was what kind of object it is. we need to know what kind of object we are dealing to know how much it is been scaled but my dear friend said no you know nothing about math lol so what do you guys say

1. xapproachesinfinity

area not volume

2. ganeshie8

wouldn't the area increase by the same factor ?

3. anonymous

suppose it is a line?

4. ganeshie8

question would be invalid then, area for a line makes no sense

5. xapproachesinfinity

that's why i said determine what object

6. anonymous

suppose it is a circle?

7. acxbox22

in the first dimension it could

8. ganeshie8

circle becomes ellipse and increasing the major axis by "k" times increases the area by "k" too so we're fine

9. xapproachesinfinity

well that's why i asked we cannot generalize to any object

10. ganeshie8

the way i see it, we can generalize... unless you have a counter example

11. acxbox22

what if it is an irregular polygon?

12. anonymous

i guess i am confused as to what the "length" of a shape it, but maybe that is just my confusion

13. ganeshie8

it doesn't matter how the shape looks

14. acxbox22

well like sat said...it depends on what length is for each shape...

15. xapproachesinfinity

i said any object and only scaling the length

16. xapproachesinfinity

well for regular gons scaling one side with some factor scales the area with the same factor

17. xapproachesinfinity

i mean not gons just rectangle, square....

18. ganeshie8

by length your friend means that he is scaling in "one" dimension

19. xapproachesinfinity

yes one dimension scaling

20. ganeshie8

imagine the object is in a paper and you "stretch" that paper horizontally by a factor of "k"

21. ganeshie8

the area scales by a factor of "k" too, no matter how the shape looks

22. acxbox22

why did u friend say that you know know nothing about math if you are right then? :P

23. xapproachesinfinity

hmm seems that way! how can we justify the general idea

24. xapproachesinfinity

well i didn't really give him an answer, i just said what kind of object are you trying to scale so i know how to answer you

25. xapproachesinfinity

but he refused and said the object matters not heheh

26. xapproachesinfinity

feel dumb now lol

27. anonymous

I am dumb

28. xapproachesinfinity

Noway! you are a mathematician, can't be dumb haha

29. xapproachesinfinity

at any rate, i think my friend and i had ego clash

30. xapproachesinfinity

@ganeshie8 can't see how we can generalize with any object? first we need to pin point what does length mean for any object

31. ganeshie8

This is not a proof, just trying to convince myself more consider below shape |dw:1432872302786:dw| stretch it horizontally by a factor of 2 |dw:1432872323889:dw|

32. ganeshie8

The claim is that the red closed shape takes twice the area of black shape

33. xapproachesinfinity

hmm i'm not really convinced there is a subtlety to how do you know it is scaled by the same factor

34. xapproachesinfinity

looks like a topology problem no?

35. ganeshie8

familiar with jacobians ?

36. ganeshie8

it is a simple change of variables problem : $X = kx\\~\\Y=y$ find the jacobian

37. xapproachesinfinity

well the word is familiar to me but i don't remember how to do it anymore haha

38. ganeshie8

jacobian gives you the scale factor for areas between the two coordinate systems which is exactly what we need

39. xapproachesinfinity

hmm i see

40. ganeshie8

$J =\begin{vmatrix} X_x&X_y\\Y_x&Y_y\end{vmatrix} = \begin{vmatrix}k&0\\0&1\end{vmatrix} = k$ therefore $dXdY = k\,dxdy$ area scales by a factor of "k" when you scale the shape in one dimension by a factor of k

41. xapproachesinfinity

Fair enough

42. xapproachesinfinity

thanks a lot!