- #1
paluskar
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I am unable to understand as to how the basis for the tangent space is
[itex]\frac{\partial}{\partial x_{i}}[/itex]. Can this be proved ,atleast intuitively?
Bachman's Forms book says that if co-ordinates of a point "p" in plane P are (x,y), then
[itex]\frac{d(x+t,y)}{dt}=\left\langle 1,0\right\rangle[/itex] [itex]\frac{d(x,y+t)}{dt}=\left\langle 0,1\right\rangle[/itex] denote vectors in [itex]\T_{p}(P)[/itex] , in fact they are the basis.
Also any point of [itex]T_{p}(P)[/itex] is [itex]dx\left\langle 0,1\right\rangle +dy\left\langle 1,0\right\rangle ;dx,dy\in\mathbb{R}[/itex]
where does this "t" come from...is it a result of parametrization??
This topic has propped up a few times in this forum, but after having gone through them I am still confused. I would appreciate any help.
I hope I am not breaking any forum rules.
[itex]\frac{\partial}{\partial x_{i}}[/itex]. Can this be proved ,atleast intuitively?
Bachman's Forms book says that if co-ordinates of a point "p" in plane P are (x,y), then
[itex]\frac{d(x+t,y)}{dt}=\left\langle 1,0\right\rangle[/itex] [itex]\frac{d(x,y+t)}{dt}=\left\langle 0,1\right\rangle[/itex] denote vectors in [itex]\T_{p}(P)[/itex] , in fact they are the basis.
Also any point of [itex]T_{p}(P)[/itex] is [itex]dx\left\langle 0,1\right\rangle +dy\left\langle 1,0\right\rangle ;dx,dy\in\mathbb{R}[/itex]
where does this "t" come from...is it a result of parametrization??
This topic has propped up a few times in this forum, but after having gone through them I am still confused. I would appreciate any help.
I hope I am not breaking any forum rules.