Vector calculus and force field

[tibphysic]

1.10.1 The force field acting on a two-dimensional linear oscillator may be described by
F=−ˆxkx − ˆyky.
Compare the work done moving against this force field when going from (1, 1) to (4, 4)
by the following straight-line paths:
(a) (1, 1)→(4, 1)→(4, 4)
(b) (1, 1)→(1, 4)→(4, 4)
(c) (1, 1)→(4, 4) along x = y.
This means evaluating
−
 (4,4)
(1,1)
F · dr
along each path.

 

[lanedance]

hi tibphysic, welcome to pf, the idea is to have a try

do you know about conservative vector fields?

 

[tibphysic]

yes i do.. conservative vectors fields does not depend on the path taken.. just on the initial and the final point.

 

[lanedance]

could be useful...

 

[LCKurtz]

1.10.1 The force field acting on a two-dimensional linear oscillator may be described by
F=−ˆxkx − ˆyky.

I'm not familiar with that notation. Is that <-kx, -ky> or something else? Are ^x and ^y notations for i and j?

 

[LCKurtz]

could be useful...

True, but perhaps the intent of the exercise is to actually work them...

 

[cragar]

im assuming his notation ^x , read x hat. where the carrot should be over the x.
is equivalent to i , j , k . they use x hat and y hat and z hat in physics more.

 

[SammyS]

For instance:

Along the path (1, 1)→(4, 1), d\hat{r}=\hat{x}\,dx\,.

So, find \displaystyle\int_{(1,1)}^{(4,1)}\,dW=\int_{x=1}^{x=4}{\vec{F}\,|_{y=1}\cdot\hat{x}}\,dx

 

[lanedance]

yeah I thought the notation was a little strange but took them as unit vectors, so the potentials were of harmonic oscillator form

 

[HallsofIvy]

For instance:

Along the path (1, 1)→(4, 1), d\hat{r}=\hat{x}\,dx\,.

So, find \displaystyle\int_{(1,1)}^{(4,1)}\,dW=\int_{x=1}^{x=4}{\vec{F}\,|_{y=1}\cdot\hat{x}}\,dx

True, but if the force field is conservative, as lanedance suggested in the first response, the problem is much easier. You don't need to do all those integrals.