The spherical substitutions $x=\rho \sin \phi \cos \theta, y=\rho \sin \phi \sin \theta$, and $z=\rho \cos \phi$ convert a smooth real-valued function $f(x, y, z)$ into a function of $\rho, \phi$, and $\theta$ : \[ w=f(\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi) . \] (a) Find formulas for $\frac{\partial w}{\partial \rho}, \frac{\partial w}{\partial \phi}$, and $\frac{\partial w}{\partial \theta}$ in terms of $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$, and $\frac{\partial f}{\partial z}$.

Step 1 ：Given the spherical substitutions $x=\rho \sin \phi \cos \theta, y=\rho \sin \phi \sin \theta$, and $z=\rho \cos \phi$ which convert a smooth real-valued function $f(x, y, z)$ into a function of $\rho, \phi$, and $\theta$ : $w=f(\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi)$.

Step 2 ：We are asked to find formulas for $\frac{\partial w}{\partial \rho}, \frac{\partial w}{\partial \phi}$, and $\frac{\partial w}{\partial \theta}$ in terms of $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$, and $\frac{\partial f}{\partial z}$.

Step 3 ：We can use the chain rule for multiple variables to find these partial derivatives. The chain rule states that the derivative of a function with respect to a variable is the sum of the derivatives of the function with respect to each of the inner variables, each multiplied by the derivative of that inner variable with respect to the outer variable.

Step 4 ：For $\frac{\partial w}{\partial \rho}$, we differentiate $w$ with respect to $\rho$, treating $\phi$ and $\theta$ as constants.

Step 5 ：For $\frac{\partial w}{\partial \phi}$, we differentiate $w$ with respect to $\phi$, treating $\rho$ and $\theta$ as constants.

Step 6 ：For $\frac{\partial w}{\partial \theta}$, we differentiate $w$ with respect to $\theta$, treating $\rho$ and $\phi$ as constants.

Step 7 ：Using the chain rule, we find: $\frac{\partial w}{\partial \rho} = \sin(\phi)\cos(\theta)\frac{\partial f}{\partial x} + \sin(\phi)\sin(\theta)\frac{\partial f}{\partial y} + \cos(\phi)\frac{\partial f}{\partial z}$

Step 8 ：$\frac{\partial w}{\partial \phi} = \rho\cos(\phi)\cos(\theta)\frac{\partial f}{\partial x} + \rho\cos(\phi)\sin(\theta)\frac{\partial f}{\partial y} - \rho\sin(\phi)\frac{\partial f}{\partial z}$

Step 9 ：$\frac{\partial w}{\partial \theta} = -\rho\sin(\phi)\sin(\theta)\frac{\partial f}{\partial x} + \rho\sin(\phi)\cos(\theta)\frac{\partial f}{\partial y}$

Step 10 ：So, the final answers are: $\boxed{\frac{\partial w}{\partial \rho} = \sin(\phi)\cos(\theta)\frac{\partial f}{\partial x} + \sin(\phi)\sin(\theta)\frac{\partial f}{\partial y} + \cos(\phi)\frac{\partial f}{\partial z}}$

Step 11 ：$\boxed{\frac{\partial w}{\partial \phi} = \rho\cos(\phi)\cos(\theta)\frac{\partial f}{\partial x} + \rho\cos(\phi)\sin(\theta)\frac{\partial f}{\partial y} - \rho\sin(\phi)\frac{\partial f}{\partial z}}$

Step 12 ：$\boxed{\frac{\partial w}{\partial \theta} = -\rho\sin(\phi)\sin(\theta)\frac{\partial f}{\partial x} + \rho\sin(\phi)\cos(\theta)\frac{\partial f}{\partial y}}$