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[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.