Question

use the fundamental theorem of line integrals to determine if F=(ysinz)i+(xsinz)j+(xycosz)k is conservative

Answer 1

when is a force conservative? is it when the gradient field is zero? can u compute the gradient field? might be m wrong or close to it!

Answer 2

where are you stuck?

Answer 3

yes, the fundamental theorem for line integrals states that if the vector field is conservative, the closed line integral is 0

Answer 4

i found the curl of F to be 0, but i wasnt sure if that theorem was that of the fundamental theorem of line integrals

Answer 5

If the the curl is 0, it so implies that the vector field is conservative

Answer 6

right

Answer 7

no wait not the curl, but the line integral...

Answer 8

\[\oint\limits_C\nabla f\cdot d\vec r=0\]

Answer 9

i am confused. does it mean that using the curl to determine if the field is conservative is not consistent with the line integral?

Answer 10

if the curl is zero that is a conservative vector field, but that doesn't apply the fundamental theorem for line integrals

Answer 11

so I'm not sure if you solved the problem the way your professor wants you to

Answer 12

i thought so too. how do i go about it?

Answer 13

I need to refresh myself on the subject... we need to show that\[\oint\limits_C\vec F\cdot d\vec r=0\]I can think of just picking some closed path and integrating it, but that wouldn't prove the general case holds...

Answer 14

or at least that\[\int_a^b\vec F\cdot d\vec r\]is path-independent

Answer 15

the problem seems a bit backwards to me; we already have the vector field, so showing that the curl is zero is the best way to show that it is conservative, but what does that have to do with using the fundamental theorem...?

Answer 16

beats my mind. i initially thought he wanted us to determine it by showing that a potential exists

Answer 17

but to show that a potential exists, i first need to find if the field is conservative

Answer 18

that's not true, we can find the potential function, though it's a pain in the butt
again that would not use the fundamental theorem though

Answer 19

example 3 here shows how to get at the potential function... takes a good minute
http://tutorial.math.lamar.edu/Classes/CalcIII/ConservativeVectorField.aspx

Answer 20

ok. i've been on this for hours. i guess i'll just use curl approach to show that the field is conservative. i cant think of any other approach

Answer 21

I'll let you know if I get any ideas, good luck!

Answer 22

thx.

Answer 23

∮C∇f⋅dr⃗ =0 if the curl is ∇f=0, then their dot product should be equal to a 0. this is how i have decided to go about it

Answer 24

sounds reasonable to me. The only other way I could think to do it would be to get the potential function and do the line integral along some random path from a to b, and show that it is invariant, but that seems a lot of work

Answer 25

i meant that u might need to compute the first partial derivatives and check if they are equal and conclude if the given function is conservative....what do u think?

Answer 26

theorem: If F = Pi + Qj is continuous in a closed and simply connected region D with first partial derivative satisfying \[\frac{ dP }{ dy}=\frac{ dQ }{ dx }, F = \Pi+Qj\] ...then F is conservative??

Answer 27

ok I dont knw if u can extend this to (i,j,k) as well but for (i,j) m sure it goes that way!

Answer 28

me too. i actually used a similar system to solve the 2d problems