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The problem is as follows:

Cartesian and polar coordinates are related by the formulas

[tex]x = r\cos\theta[/tex]

[tex]y = r\sin\theta[/tex]

Determine [tex]\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial\theta}{\partial x}, and \frac{\partial\theta}{\partial x}[/tex]. Differentiate the equations above implcitly adn then solve the resulting system of 4 equations in the four unknowns [tex]\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial\theta}{\partial x}, and \frac{\partial\theta}{\partial x}[/tex].

Now, this seems a whole lot harder than just differentiating [tex]r^2 = x^2 + y^2[/tex], but I did anyway and the 4 equations are:

[tex]\frac{\partial x}{\partial x} = \frac{\partial r}{\partial x}\cos\theta - \sin\theta\frac{\partial\theta}{\partial x}r[/tex]

[tex]\frac{\partial x}{\partial y} = \frac{\partial r}{\partial y}\cos\theta - \sin\theta\frac{\partial\theta}{\partial y}r[/tex]

[tex]\frac{\partial y}{\partial x} = \frac{\partial r}{\partial x}\sin\theta + \cos\theta\frac{\partial\theta}{\partial x}r[/tex]

[tex]\frac{\partial y}{\partial y} = \frac{\partial r}{\partial y}\sin\theta + \cos\theta\frac{\partial\theta}{\partial y}r[/tex]

Now, if I try to solve for [tex]\frac{\partial r}{\partial x}[/tex], I know what the answer should be by differentiating [tex]r^2 = x^2 + y^2[/tex], and it's [tex]\frac xr[/tex] or [tex]\cos\theta[/tex], however I can't seem to get that from combining the 4 equations.

What am I doing wrong?

Thanks in advance.

Cartesian and polar coordinates are related by the formulas

[tex]x = r\cos\theta[/tex]

[tex]y = r\sin\theta[/tex]

Determine [tex]\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial\theta}{\partial x}, and \frac{\partial\theta}{\partial x}[/tex]. Differentiate the equations above implcitly adn then solve the resulting system of 4 equations in the four unknowns [tex]\frac{\partial r}{\partial x}, \frac{\partial r}{\partial y}, \frac{\partial\theta}{\partial x}, and \frac{\partial\theta}{\partial x}[/tex].

Now, this seems a whole lot harder than just differentiating [tex]r^2 = x^2 + y^2[/tex], but I did anyway and the 4 equations are:

[tex]\frac{\partial x}{\partial x} = \frac{\partial r}{\partial x}\cos\theta - \sin\theta\frac{\partial\theta}{\partial x}r[/tex]

[tex]\frac{\partial x}{\partial y} = \frac{\partial r}{\partial y}\cos\theta - \sin\theta\frac{\partial\theta}{\partial y}r[/tex]

[tex]\frac{\partial y}{\partial x} = \frac{\partial r}{\partial x}\sin\theta + \cos\theta\frac{\partial\theta}{\partial x}r[/tex]

[tex]\frac{\partial y}{\partial y} = \frac{\partial r}{\partial y}\sin\theta + \cos\theta\frac{\partial\theta}{\partial y}r[/tex]

Now, if I try to solve for [tex]\frac{\partial r}{\partial x}[/tex], I know what the answer should be by differentiating [tex]r^2 = x^2 + y^2[/tex], and it's [tex]\frac xr[/tex] or [tex]\cos\theta[/tex], however I can't seem to get that from combining the 4 equations.

What am I doing wrong?

Thanks in advance.

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