- #1
spaghetti3451
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Consider a curve ##C:{\bf{x}}={\bf{F}}(t)##, for ##a\leq t \leq b##, in ##\mathbb{R}^{3}## (with ##x## any coordinates). oriented so that ##\displaystyle{\frac{d}{dt}}## defines the positive orientation in ##U=\mathbb{R}^{1}##. If ##\alpha^{1}=a_{1}dx^{1}+a_{2}dx^{2}+a_{3}dx^{3}## is a ##1##-form on ##\mathbb{R}^{3}##, then its integral or line integral over ##C## becomes
##\displaystyle{\int_{C}\ \alpha^{1} = \int_{C}\ \sum\limits_{i}a_{i}(x)dx^{i}}##
##\displaystyle{=\int_{a}^{b} F^{*}\bigg[\sum\limits_{i}a_{i}(x)dx^{i}\bigg]}##
##\displaystyle{=\int_{a}^{b} F^{*}\bigg[\sum\limits_{i}a_{i}(x(t))\frac{dx^{i}}{dt}\bigg]dt}##
Thus ##\displaystyle{\int_{(U,o;F)}\alpha^{p}:=\int_{(U,o)}F^{*}\alpha^{p}}## is the usual rule for evaluating a line integral over an oriented parameterized curve! We may write this as
##\displaystyle{\int_{C}\alpha^{1}=\int_{a}^{b}\alpha^{1}\left(\frac{d{\bf{x}}}{dt}\right)dt}##
and so the integral of a ##1##-form over an oriented parameterized curve ##C## is simply the ordinary integral of the function that assigns to the parameter ##t## the value of the ##1##-form on the velocity vector at ##{\bf{x}}(t)##.
What does it mean for ##\displaystyle{\frac{d}{dt}}## to define the positive orientation in ##U=\mathbb{R}^{1}##
##\displaystyle{\int_{C}\ \alpha^{1} = \int_{C}\ \sum\limits_{i}a_{i}(x)dx^{i}}##
##\displaystyle{=\int_{a}^{b} F^{*}\bigg[\sum\limits_{i}a_{i}(x)dx^{i}\bigg]}##
##\displaystyle{=\int_{a}^{b} F^{*}\bigg[\sum\limits_{i}a_{i}(x(t))\frac{dx^{i}}{dt}\bigg]dt}##
Thus ##\displaystyle{\int_{(U,o;F)}\alpha^{p}:=\int_{(U,o)}F^{*}\alpha^{p}}## is the usual rule for evaluating a line integral over an oriented parameterized curve! We may write this as
##\displaystyle{\int_{C}\alpha^{1}=\int_{a}^{b}\alpha^{1}\left(\frac{d{\bf{x}}}{dt}\right)dt}##
and so the integral of a ##1##-form over an oriented parameterized curve ##C## is simply the ordinary integral of the function that assigns to the parameter ##t## the value of the ##1##-form on the velocity vector at ##{\bf{x}}(t)##.
What does it mean for ##\displaystyle{\frac{d}{dt}}## to define the positive orientation in ##U=\mathbb{R}^{1}##