# Vector calculus and force field

1. Jun 29, 2011

### 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jun 29, 2011

### lanedance

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

do you know about conservative vector fields?

3. Jul 1, 2011

### tibphysic

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

4. Jul 1, 2011

### lanedance

could be useful....

5. Jul 1, 2011

### LCKurtz

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

6. Jul 1, 2011

### LCKurtz

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

7. Jul 1, 2011

### 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.

8. Jul 1, 2011

### SammyS

Staff Emeritus
For instance:

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

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

Last edited: Jul 1, 2011
9. Jul 1, 2011

### lanedance

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

10. Jul 2, 2011

### HallsofIvy

Staff Emeritus
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.