Compute the work done by the force field.

[bfusco]

Homework Statement

compute the work done by the force field F(xyz)=[(yze^(xyz)+y^2+1)i+(xze^(xyz)+2xy)j+(xye^(xyz))k], in moving the object along the path C from beginning to end. here C is the path paramterized by r(t)=<t, (t^2)-1, t+2>, 0<_t<_1.

The Attempt at a Solution

Would i be correct if i said that i should use the functions of x(t), y(t), z(t), and substituting them into the original equation? then take the derivative of r(t) and do the dot product of them. then use the formula: ∫F*dr?

 

[LCKurtz]

Homework Statement

compute the work done by the force field F(xyz)=[(yze^(xyz)+y^2+1)i+(xze^(xyz)+2xy)j+(xye^(xyz))k], in moving the object along the path C from beginning to end. here C is the path paramterized by r(t)=<t, (t^2)-1, t+2>, 0<_t<_1.

The Attempt at a Solution

Would i be correct if i said that i should use the functions of x(t), y(t), z(t), and substituting them into the original equation? then take the derivative of r(t) and do the dot product of them. then use the formula: ∫F*dr?

If you mean$$
\int_0^1\vec F\cdot \frac{d\vec R}{dt}\, dt$$then yes.

 

[haruspex]

Would i be correct if i said that i should use the functions of x(t), y(t), z(t), and substituting them into the original equation? then take the derivative of r(t) and do the dot product of them. then use the formula: ∫F*dr?

Some of that sounds right, other parts bewildering. To make it clearer, try it and post your working.

 

[HallsofIvy]

If you mean$$
\int_0^1\vec F\cdot \frac{d\vec R}{dt}\, dt$$then yes.

Actually, given that $\vec{r}(t)$ is a function of t, "$d\vec{r}$" is a reasonable way of writing "$(d\vec{r}/dt) dt$". But I would object to using the capital R when only "r" was given before.

 

[LCKurtz]

Actually, given that $\vec{r}(t)$ is a function of t, "$d\vec{r}$" is a reasonable way of writing "$(d\vec{r}/dt) dt$". But I would object to using the capital R when only "r" was given before.

Yeah, I used the capital letter out of habit. But my point was to make sure the OP understood the independent variable was $t$ and the use of the $t$ limits.