Question

Find the integral along a path s between two points (0,0) and (1,1) of the component of the velocity field u in the direction of s for the following three cases:

a. s is a straight line
b. s is a parabola with vertex at the origin and opening upwards,
c. s is a portion of hte y-axis and a straight line perpendicular to it.

the velocity field u is given by u= [ u, v]
where u = x^2+y^2 and v = xy^2

Answer 1

Hey man, maybe we can get the ball rolling. What're you having trouble with?

Answer 2

So i think i've got this right. the answer i get is u = [2/3 , 1/6] for a. Its just that it's been maybe 2 years since i've done double integrals and i'm having a hard time with setting this problem up.

Answer 3

or well not so much a hard time but that i'm unsure if it is correct.

Answer 4

Ha, well same here. So the straight line would a constant for y

Answer 5

Well, or x

Answer 6

Here is how i did it. since it is a straight line y=ax+b.

Now the integral looks like: \[\int\limits_{0}^{1} \int\limits_{0}^{1} <x^2 + y^2 > <x^2y> dxdy\]

This means i need to do two integrals. One looks like:

\[\int\limits_{0}^{1} \int\limits_{0}^{1} x^2 + y^2 dxdy\]

the other looks like:

\[\int\limits_{0}^{1} \int\limits_{0}^{1} x^2y dxdy\]

Is this right?

Answer 7

x is u i guess and since u is a matrix with two components...

Answer 8

Yes, I believe you do need to do a double integral. So if you do do this the way you propose, double integrating on each function, you have found a value for each variable [u,v] along the path s

Answer 9

yeah thats what i was thinking... but its tough when you have no feedback. I hate turning in things to my professor when i'm not positive I have it correct

Answer 10

Also, no one i know im my class is even at this problem yet.

Answer 11

So if next you do a parabola, assume y=x^2, where would you place the path function?

Answer 12

the velocity field u would be inserted into x. so you would have two integrals again

Answer 13

same as before but were now squaring the two functions of u and v

Answer 14

All right, and for the other path which is a step function of sorts, part of y=VAR and x = Var

Answer 15

Where they intersect in the space [0,1]

Answer 16

yeah looks like we have two equations and now i have to do four equations....

Answer 17

piecewise equation business going on here. lol

Answer 18

Yep, so this is your physic work?

Answer 19

No. for a class called Aerodynamics of Aerovehicles. This homework is just math warmup for what's to come. This was the 2nd problem set and the first one that really uses any math.

Answer 20

Oh, nice!

Answer 21

its been three years for me since calculus three, three years since dynamics, and two years ago i might have done some of this in fluid mechaics but not much. So its very difficult for me. haha. Thanks for looking at my work.

Answer 22

Hey, sorry, but I've got to leave to have some dinner. I need to brush up on my Calc skills because helping out on these more advanced problems is quite fun. But I think you're less rusty than I am! Good luck with the work! I'll check back on the site, there are some solid helpers here

Answer 23

Haha first time i found this site. I really like it. Its a great idea. I might have to come on here sometime when i have free time and help others too. Thanks again. I'll give ya a medal.

Answer 24

How'd you find the site?

Answer 25

an advertisement on a webpage that has a bunch of multivariate calculus stuff.

Answer 26

Cool. Thanks. See you around

Answer 27

c you.