Is the transformation from Cartesian to Spherical polar coordinates an orthogonal transformation?

Question

I have been trying to understand Tensors. I want to know if transformation between two different curvi-linear coordinates is an orthogonal transformation?

I am getting the answer by some analysis that it is not an orthogonal transformation.

Answer 1

This is a long comment:

The notion of an orthogonal transformations only applies to linear (or affine maps). So if this is what you really mean, then the answer is "you're correct, this is not an orthogonal transformation, because it's not even an affine transformation."

However, I think that what you're really looking for is the notion of a conformal transformation. By definition, these are functions are preserves angles locally.

Quoting this:

A remarkable geometrical property enjoyed by all complex analytic functions is that, at non-critical points, they preserve angles, and therefore define conformal mappings.

So the complex exponential map $\mathbb{C} \rightarrow \mathbb{C}$, which has no critical points, must therefore be conformal. You can think of this as a map from polar coordinates to cartesian coordinates; the real part of the input tells you $e$ to the power of the radius, and the imaginary part tells you the angle from the $x$-axis, anticlockwise, in radians.

However this, doesn't answer your question. In particular, you're probably more interested in the function $f : \mathbb{C}_{\Re > 0} \rightarrow \mathbb{C}$ defined by $f(z)=\Re z\cdot \frac{e^z}{e^{\Re z}},$ since the distance from $f(z)$ to the origin is exactly $\Re z$ in this case. I don't know if $f$ is conformal or not, but I've made this "answer" community wiki, so if anyone knows, they can edit it.