The evaluation of the pullback of a constant 1-form \( k_{1}dx + k_{2}dy + k_{3}dz \) over a directed line segment from point \(\bold{r}\) to point \(\bold{s}\) is independent of the linear parameterization used. The parameterization is expressed as \((x,y,z) = \bold{r} + t(\bold{s}-\bold{r})\). The coefficients of the 1-form remain constant regardless of the chosen parameterization, confirming that the pullback's evaluation is consistent across different linear mappings.
PREREQUISITESMathematicians, physics students, and anyone studying differential geometry or vector calculus who seeks to understand the behavior of constant 1-forms under different parameterizations.
tronter said:Show that the evaluation of the pullback of a constant 1-form [tex]k_{1}dx + k_{2}dy + k_{3}dz[/tex] over the directed line segment from [tex]\bold{r}[/tex] to [tex]\bold{s}[/tex] does not depend on which linear parameterization is chosen.
So [tex](x,y,z) = \bold{r} + t(\bold{s}-\bold{r})[/tex]. Then what?