- #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
- Start date
In summary, the conversation is about a change of coordinates from rectangular to polar and the correct way to differentiate certain equations involving x and y. The correct expressions for differentiating y with respect to theta and x with respect to r are provided, along with the suggestion to start from x = r cos theta and y = r sin theta when finding derivatives with respect to r and theta.
Change of coordinates from rectangular (x,y) to polar (r,θ). Not sure what's wrong with my working..
- #2
pasmith
Science Advisor
Homework Helper
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Your error is that you have differentiated [itex]y = x \tan\theta[/itex] incorrectly. You should get
[tex]\frac{\partial y}{\partial \theta} = x\sec^2\theta + \frac{\partial x}{\partial \theta}\tan\theta[/tex]
because [itex]x[/itex] is also a function of [itex]\theta[/itex].
Also, to work out [itex]\partial x/\partial r[/itex], you would need to differentiate
[tex]r^2 = x^2 + y^2[/tex]
with respect to [itex]r[/itex] with [itex]\theta[/itex] held constant, which again gives
[tex]2r = 2x\frac{\partial x}{\partial r} + 2y\frac{\partial y}{\partial r}[/tex]
because y is also a function of [itex]r[/itex].
If you're trying to find derivatives with respect to r and [itex]\theta[/itex], it's best to start from [itex]x = r\cos\theta[/itex], [itex]y = r\sin\theta[/itex].
[tex]\frac{\partial y}{\partial \theta} = x\sec^2\theta + \frac{\partial x}{\partial \theta}\tan\theta[/tex]
because [itex]x[/itex] is also a function of [itex]\theta[/itex].
Also, to work out [itex]\partial x/\partial r[/itex], you would need to differentiate
[tex]r^2 = x^2 + y^2[/tex]
with respect to [itex]r[/itex] with [itex]\theta[/itex] held constant, which again gives
[tex]2r = 2x\frac{\partial x}{\partial r} + 2y\frac{\partial y}{\partial r}[/tex]
because y is also a function of [itex]r[/itex].
If you're trying to find derivatives with respect to r and [itex]\theta[/itex], it's best to start from [itex]x = r\cos\theta[/itex], [itex]y = r\sin\theta[/itex].
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FAQ: What's wrong with my Jacobian of polar coordinates?
What is the Jacobian of polar coordinates?
The Jacobian of polar coordinates is a mathematical concept used to calculate the change in variables when converting from Cartesian coordinates to polar coordinates. It is a 2x2 matrix that represents the partial derivatives of the polar coordinates with respect to the Cartesian coordinates.
How is the Jacobian of polar coordinates calculated?
The Jacobian of polar coordinates can be calculated using the determinant of the matrix, which is equal to the partial derivative of the polar coordinate r with respect to x multiplied by the partial derivative of the polar coordinate θ with respect to y, minus the partial derivative of r with respect to y multiplied by the partial derivative of θ with respect to x.
What is the significance of the Jacobian of polar coordinates?
The Jacobian of polar coordinates is important in many fields of science and engineering, as it allows for the conversion of equations and calculations from Cartesian coordinates to polar coordinates. It is also used in transformations and integrals to solve problems in a more efficient way.
How does the Jacobian of polar coordinates relate to other coordinate systems?
The Jacobian of polar coordinates is just one example of a Jacobian matrix, which is used to convert between different coordinate systems. Other examples include the Jacobian of cylindrical coordinates and the Jacobian of spherical coordinates. They all follow a similar calculation process and serve the same purpose of transforming equations and calculations between coordinate systems.
What are some practical applications of the Jacobian of polar coordinates?
The Jacobian of polar coordinates is used in many fields of science and engineering, such as physics, mathematics, and robotics. It is especially useful in solving problems involving circular and rotational motion, as well as in the analysis of electromagnetic fields and fluid dynamics.