Does the Parameterization Affect the Pullback of a Constant 1-Form?

[tronter]

Show that the evaluation of the pullback of a constant 1-form $$k_{1}dx + k_{2}dy + k_{3}dz$$ over the directed line segment from $$\bold{r}$$ to $$\bold{s}$$ does not depend on which linear parameterization is chosen.

So $$(x,y,z) = \bold{r} + t(\bold{s}-\bold{r})$$. Then what?

 

[Reb]

Show that the evaluation of the pullback of a constant 1-form $$k_{1}dx + k_{2}dy + k_{3}dz$$ over the directed line segment from $$\bold{r}$$ to $$\bold{s}$$ does not depend on which linear parameterization is chosen.

So $$(x,y,z) = \bold{r} + t(\bold{s}-\bold{r})$$. Then what?

Τhen, you observe that the coefficients of the 1-form are constant, and will remain so
no matter what (x,y,z) you choose.


Ps. Yeap, I am answering old questions.
Got time on my hands and need to kill it... Is that so bad?