- #1
unscientific
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Homework Statement
Change of coordinates from rectangular (x,y) to polar (r,θ). Not sure what's wrong with my working..
What's wrong with my Jacobian of polar coordinates?
- Thread starter unscientific
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The discussion focuses on correcting the differentiation of polar coordinates, specifically the transformation from rectangular coordinates (x, y) to polar coordinates (r, θ). The main error identified is in the differentiation of y = x tan(θ), where it should include the derivative of x with respect to θ. Additionally, when finding ∂x/∂r, the equation r² = x² + y² must be differentiated with respect to r while holding θ constant. It is recommended to start from the equations x = r cos(θ) and y = r sin(θ) for clarity in deriving the necessary derivatives. Proper differentiation is crucial for accurately calculating the Jacobian in polar coordinates.
Change of coordinates from rectangular (x,y) to polar (r,θ). Not sure what's wrong with my working..
- #2
pasmith
Science Advisor
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Your error is that you have differentiated y = x \tan\theta incorrectly. You should get
\frac{\partial y}{\partial \theta} = x\sec^2\theta + \frac{\partial x}{\partial \theta}\tan\theta
because x is also a function of \theta.
Also, to work out \partial x/\partial r, you would need to differentiate
r^2 = x^2 + y^2
with respect to r with \theta held constant, which again gives
2r = 2x\frac{\partial x}{\partial r} + 2y\frac{\partial y}{\partial r}
because y is also a function of r.
If you're trying to find derivatives with respect to r and \theta, it's best to start from x = r\cos\theta, y = r\sin\theta.
\frac{\partial y}{\partial \theta} = x\sec^2\theta + \frac{\partial x}{\partial \theta}\tan\theta
because x is also a function of \theta.
Also, to work out \partial x/\partial r, you would need to differentiate
r^2 = x^2 + y^2
with respect to r with \theta held constant, which again gives
2r = 2x\frac{\partial x}{\partial r} + 2y\frac{\partial y}{\partial r}
because y is also a function of r.
If you're trying to find derivatives with respect to r and \theta, it's best to start from x = r\cos\theta, y = r\sin\theta.
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