送交者: antonin 于 2006-2-01, 15:38:13:
回答: Would anyone please recommend a book on geometric meanings 由 mangolasi 于 2006-1-31, 15:27:48:
I have not seen any book specifically on the geometric interpretations of things. I believe when you learn something, you want to understand what it really is, like the determinant, if you don't just look at the formal definitions (as for most concepts in Math). Also, not everything can be explained geometrically. I believe most of time, familiarity is the key in understanding. "A lie repeated a thousand times turned out to be a truth":) Back to your determinant in the change of variable formula, if the transformation is linear, then you need only to multiply by the determinant of the transformation. In the general case, if the transformation and domain of integration is "nice" enough, then locally you can approximate the transformation by a linear one, which is its derivative(a linear mapping whose matrix is the Jacobian matrix), then you can do the same locally. Now for the whole domain, you cut it into small enough pieces, use the linear approximation for each piece, then add up, which should give you a itegral sum(Riemann sum). Taking the limit may get rid of the erorr and give you the integral, hence the formula.