DIFFERENTIAL FORMS Differentials Are an Important Topic in Math Ucdavis

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here we're continuing our path down a very elementary approach to differential forms and just recall where we came from so last time we determined the difference between RN and T PRN in other words the tangent space based at P of RN and the idea was that RN were these points in n space and then the elements from the tangent space at P of R in other in other words T PR in those were n dimensional vectors based at a point P in RN so I've drawn a little picture here so R 2 vs T PR 2 so R 2 is just this plane notice I've got the x-axis and the y-axis so generally we'll take x and y to be these coordinate functions that project to these axes and then what we have here is this point P which I've put in yellow in r2 so that point is just sitting there and then I pick a new coordinate system at that point P and that new coordinate system has axes that are orthogonal to each other DX and dy and then this vector VP is inside TP r2 and so notice it's a vector that is based at P and kind of point