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Mathematics
Differential Geometry
Infinitesimal area element in polar coordinate
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[QUOTE="PeroK, post: 5462020, member: 493650"] This is really about a change of variables, rather than comparing a specific small area in two coordinate systems. As you have seen, if you have a small rectangle defined by ##dx, dy## and the area of this rectangle is ##dA## then ##dA \ne rdrd\theta##. In fact, it's not even clear what ##dr## and ##d\theta## would be in this case. To see this, draw the rectangle and try to figure out what ##dr, d\theta## would be. Instead, if you imagine integrating in polar coordinates, you will take the limit of small areas, defined by ##dr, d\theta##. These are not rectangles (although they are close). Each has an area of approximately ##dA_{p} = rd\theta dr ##. This leads to: ##\int dxdy = \int rdrd\theta## But, there is not a direct relationship between the area of a specific small rectangular area in Cartesian coordinates and a different specific small (almost) rectangular area in polar coordinates. [/QUOTE]
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Mathematics
Differential Geometry
Infinitesimal area element in polar coordinate
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