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What is the application of $(f_x)^2 + (f_y)^2 = (g_r)^2 + \frac{1}{r^2} (g_\theta)^2$?

$f(x, y)$ is a smooth 2 variable function and $g(r, \theta) := f(r \cos \theta, r \sin \theta)$.

I can understand the above equation is true, but I don't know why we are glad to know the above equation.

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    Without any context this is a strange question. Why are you asking about this in particular? You can ask this about any random identity and in most cases there will not be some amazing application for it. It is just a useful fact. In particular if you work with PDEs then knowing various identities that relates gradients (like here), Laplacians and similar operators in different coordinate systems will make life easier for you. However it's often no need to memorize it as long as you remember how to derive it when you need it. – 2017-02-13
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    I am reading a calculus book, and the author says that this equation is very useful. So I asked the above question. The author says that $f_{xx} + f_{yy} = g_{rr} + \frac{1}{r} g_{r} + \frac{1}{r^2} g_{\theta \theta}$ is also very useful. – 2017-02-13
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    Yes, that is the so-called Laplacian which I mentioned above in polar coordinates. A large set of differential equations involve Laplacians and gradients so for example when solving these and the domain we want to solve it over and/or the initial/boundary conditions have polar symmetry (e.g. solving the equation on a disc) then it makes life much easier if you change to polar coordinates (for which you need to derive it or remember this expression). – 2017-02-13
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    As for some concrete examples where this is useful (sticking to the Laplacian as it's easier to give good examples): in physics the Laplace operator pops up many places like for computing [electric fields](https://en.wikipedia.org/wiki/Laplace's_equation#Electrostatics), for computing [gravitational fields](https://en.wikipedia.org/wiki/Poisson's_equation#Newtonian_gravity), [fluid flow](https://en.wikipedia.org/wiki/Laplace's_equation#Fluid_flow) ... In many many of these physical cases we will have polar/spherical symmetry making these expressions useful. – 2017-02-13
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    Thank you very much, Mr. Winther. I will see a book of electromagnetics. – 2017-02-13

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