Prove columns of Jacobian Matrix of $T(r,\varphi,\theta) := (r\ \cos \varphi \cos \theta ,\ r\ \sin \varphi \cos \theta ,r\ \sin \theta)$ orthogonal

Question

Let $f:\mathbb{R^3} \to \mathbb{R}$ be a differentiable function.

For $r > 0, \varphi \in [0,2\pi]$ and $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ we look at the functions

$$T(r,\varphi,\theta) := (r\ \cos \varphi \cos \theta ,\ r\ \sin \varphi \cos \theta ,r\ \sin \theta)$$ and

$$g(r,\varphi, \theta) = f(T(r,\varphi, \theta))$$

How can one prove that the columns of the Jacobian Matrix of $T$ are orthogonal perpendicular to one another?

For which values of $r,\varphi,\theta$ is $T'$ regular and how do $\nabla f$ and $\nabla g$ calculate into each other?

I know that Laplace's equation is given by

$$\nabla^2u = u_{xx}+u_{yy}+u_{zz} = 0$$

and in another thread Laplace's equation in spherical coordinates is proven.

I found this on the internet:

but I still don't know what exactly needs to be done to prove that the columns of the Jacobian Matrix of T are orthogonal (perpendicular to one another) and how to go on with the other question..

Answer 1

Since $T(r,\varphi,\theta) = (r\cos\varphi\cos\theta,r\sin\varphi\cos\theta,r\sin\theta)$,

$$JT_{(r,\varphi,\theta)} =\left(\begin{array}{ccc} \cos\varphi\cos\theta & -r\sin\varphi\cos\theta & -r\cos\varphi\sin\theta\\ \sin\varphi\cos\theta & r\cos\varphi\cos\theta & -r\sin\varphi\sin\theta\\ \sin\theta & 0 & r\cos\theta \end{array}\right) = (x_1\quad x_2\quad x_n).$$

Now, check $x_i \cdot x_j$.

Answer 2

Arrange the derivatives $\frac{\partial T}{\partial r}$,$\frac{\partial T}{\partial \varphi}$ and $\frac{\partial T}{\partial \theta}$ as columns and do the inner product among them.