Christina DanielI would like to evaluate two more derivatives. They are $ \frac{dx}{d\phi}\big|_{\phi^+} $ and $ \frac{dz}{d\theta}\big|_{\theta^+} $ given $x = r \sin\theta\cos\phi$ and $z = r\cos\theta$.
Start with $ \frac{dx}{d\phi}\big|_{\phi^+} $. The first step is to substitute $x$ with $r \sin\theta \cos\phi$.
$ \frac{dx}{d\phi}\big|_{\phi^+} = \frac{d r \sin\theta \cos \phi}{d\phi}\big|_{\phi^+} $.
The next step is to apply the product rule to the right side.
$\frac{dx}{d\phi}\big|_{\phi^+} = r \sin\theta \frac{d \cos \phi}{d\phi}\big|_{\phi^+} + \cos \phi \frac{dr \sin\theta}{d\phi}\big|_{\phi^+}$.
Using this derivative of cosine, the previous equation becomes
$\frac{dx}{d\phi}\big|_{\phi^+} = -r \sin\theta \sin\phi + \cos \phi \frac{d r \sin\theta}{d\phi}\big|_{\phi^+}$.
From this post about the derivative of a constant function,
$ \boxed{ \frac{dx}{d\phi}\bigg|_{\phi^+} = -r \sin\theta \sin\phi }$.
Second, evaluate $ \frac{dz}{d\theta}\big|_{\theta^+} $. The first step is to substitute $z$ with $r \cos\theta$.
$ \frac{dz}{d\theta}\big|_{\theta^+} = \frac{dr\cos\theta }{d\theta}\big|_{\theta^+} $.
The next step is to apply the product rule to the right side.
$ \frac{dz}{d\theta}\big|_{\theta^+} = r \frac{d \cos\theta}{d\theta}\big|_{\theta^+} + \cos\theta \frac{dr}{d\theta} \big|_{\theta^+}$.
Using this derivative of cosine, the previous equation becomes
$ \frac{dz}{d\theta}\big|_{\theta^+} = -r \sin\theta + \cos\theta \frac{dr}{d\theta} \big|_{\theta^+}$.
From this post about the derivative of a constant function,
$ \boxed{ \frac{dz}{d\theta}\bigg|_{\theta^+} = -r \sin\theta }$.
And with that, all nine derivatives of this intimidating matrix have been determined.
