10. for the specialized coordinate system given, express the derivatives of a general function f with…

10. for the specialized coordinate system given, express the derivatives of a general function f with respect to the new variables in terms of (\frac{partial f}{partial x},\frac{partial f}{partial y},\frac{partial f}{partial z}). (a) cylindrical coordinates: ((r,\theta,z)). cylindrical coordinates are given by: (x = rcos\theta,y = rsin\theta,z = z), and/or (r=sqrt{x^{2}+y^{2}},\theta=arctan(y / x),z = z). (b) spherical coordinates: ((\rho,\theta,phi)). spherical coordinates are given by: (x=\rhocos\thetasinphi,y = \rhosin\thetasinphi,z=\rhocosphi), and/or (\rho=sqrt{x^{2}+y^{2}+z^{2}},\theta=arctan(y / x),phi=arccosleft(\frac{z}{sqrt{x^{2}+y^{2}+z^{2}}}\right))

10. for the specialized coordinate system given, express the derivatives of a general function f with respect to the new variables in terms of (\frac{partial f}{partial x},\frac{partial f}{partial y},\frac{partial f}{partial z}). (a) cylindrical coordinates: ((r,\theta,z)). cylindrical coordinates are given by: (x = rcos\theta,y = rsin\theta,z = z), and/or (r=sqrt{x^{2}+y^{2}},\theta=arctan(y / x),z = z). (b) spherical coordinates: ((\rho,\theta,phi)). spherical coordinates are given by: (x=\rhocos\thetasinphi,y = \rhosin\thetasinphi,z=\rhocosphi), and/or (\rho=sqrt{x^{2}+y^{2}+z^{2}},\theta=arctan(y / x),phi=arccosleft(\frac{z}{sqrt{x^{2}+y^{2}+z^{2}}}\right))

Answer

Explanation:

Step1: Recall the chain - rule for partial derivatives

The chain - rule for partial derivatives states that if (f(x,y,z)) and (x = x(u,v,w)), (y = y(u,v,w)), (z = z(u,v,w)), then (\frac{\partial f}{\partial u}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial u}).

(a) Cylindrical coordinates ((r,\theta,z)) where (x = r\cos\theta), (y = r\sin\theta), (z = z)

Step2: Calculate (\frac{\partial f}{\partial r})

Using the chain - rule (\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial r}). Since (\frac{\partial x}{\partial r}=\cos\theta), (\frac{\partial y}{\partial r}=\sin\theta), (\frac{\partial z}{\partial r} = 0), we have (\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\cos\theta+\frac{\partial f}{\partial y}\sin\theta).

Step3: Calculate (\frac{\partial f}{\partial\theta})

Using the chain - rule (\frac{\partial f}{\partial\theta}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial\theta}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial\theta}). Since (\frac{\partial x}{\partial\theta}=-r\sin\theta), (\frac{\partial y}{\partial\theta}=r\cos\theta), (\frac{\partial z}{\partial\theta} = 0), we have (\frac{\partial f}{\partial\theta}=-r\frac{\partial f}{\partial x}\sin\theta + r\frac{\partial f}{\partial y}\cos\theta).

Step4: Calculate (\frac{\partial f}{\partial z})

Since (z = z), (\frac{\partial f}{\partial z}=\frac{\partial f}{\partial z}) (no change in the transformation for (z)).

(b) Spherical coordinates ((\rho,\theta,\phi)) where (x=\rho\cos\theta\sin\phi), (y = \rho\sin\theta\sin\phi), (z=\rho\cos\phi)

Step5: Calculate (\frac{\partial f}{\partial\rho})

Using the chain - rule (\frac{\partial f}{\partial\rho}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial\rho}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial\rho}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial\rho}). Since (\frac{\partial x}{\partial\rho}=\cos\theta\sin\phi), (\frac{\partial y}{\partial\rho}=\sin\theta\sin\phi), (\frac{\partial z}{\partial\rho}=\cos\phi), we have (\frac{\partial f}{\partial\rho}=\frac{\partial f}{\partial x}\cos\theta\sin\phi+\frac{\partial f}{\partial y}\sin\theta\sin\phi+\frac{\partial f}{\partial z}\cos\phi).

Step6: Calculate (\frac{\partial f}{\partial\theta})

Using the chain - rule (\frac{\partial f}{\partial\theta}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial\theta}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial\theta}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial\theta}). Since (\frac{\partial x}{\partial\theta}=-\rho\sin\theta\sin\phi), (\frac{\partial y}{\partial\theta}=\rho\cos\theta\sin\phi), (\frac{\partial z}{\partial\theta}=0), we have (\frac{\partial f}{\partial\theta}=-\rho\frac{\partial f}{\partial x}\sin\theta\sin\phi+\rho\frac{\partial f}{\partial y}\cos\theta\sin\phi).

Step7: Calculate (\frac{\partial f}{\partial\phi})

Using the chain - rule (\frac{\partial f}{\partial\phi}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial\phi}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial\phi}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial\phi}). Since (\frac{\partial x}{\partial\phi}=\rho\cos\theta\cos\phi), (\frac{\partial y}{\partial\phi}=\rho\sin\theta\cos\phi), (\frac{\partial z}{\partial\phi}=-\rho\sin\phi), we have (\frac{\partial f}{\partial\phi}=\frac{\partial f}{\partial x}\rho\cos\theta\cos\phi+\frac{\partial f}{\partial y}\rho\sin\theta\cos\phi-\frac{\partial f}{\partial z}\rho\sin\phi).

Answer:

(a) Cylindrical coordinates

(\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\cos\theta+\frac{\partial f}{\partial y}\sin\theta), (\frac{\partial f}{\partial\theta}=-r\frac{\partial f}{\partial x}\sin\theta + r\frac{\partial f}{\partial y}\cos\theta), (\frac{\partial f}{\partial z}=\frac{\partial f}{\partial z})

(b) Spherical coordinates

(\frac{\partial f}{\partial\rho}=\frac{\partial f}{\partial x}\cos\theta\sin\phi+\frac{\partial f}{\partial y}\sin\theta\sin\phi+\frac{\partial f}{\partial z}\cos\phi), (\frac{\partial f}{\partial\theta}=-\rho\frac{\partial f}{\partial x}\sin\theta\sin\phi+\rho\frac{\partial f}{\partial y}\cos\theta\sin\phi), (\frac{\partial f}{\partial\phi}=\frac{\partial f}{\partial x}\rho\cos\theta\cos\phi+\frac{\partial f}{\partial y}\rho\sin\theta\cos\phi-\frac{\partial f}{\partial z}\rho\sin\phi)