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If we expres cartesian cordinates in polar coordinates we get:

x=r*cos(theta)

y=r*sin(theta)

let's differentiate those 2 eqs:

dx= dr cos(theta) -r* d(theta) * sin(theta)

dy=dr sin(theta) + r* d(theta) * cos(theta)

why isn't dx*dy= r* dr* d(theta) ( like when taking the jacobian , or when doing geometric interpretation)?

x=r*cos(theta)

y=r*sin(theta)

let's differentiate those 2 eqs:

dx= dr cos(theta) -r* d(theta) * sin(theta)

dy=dr sin(theta) + r* d(theta) * cos(theta)

why isn't dx*dy= r* dr* d(theta) ( like when taking the jacobian , or when doing geometric interpretation)?

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