We can expand a smooth function $f:\mathbb{R}^3\to \mathbb{R}$ in a Taylor series: $$f((x^1,x^2,x^3)+(h^1,h^2,h^3))=f(x^1,x^2,x^3)+h_i\frac{\partial f}{\partial x^i}+h_ih_j\frac{\partial^2 f}{\partial x^i\,\partial x^j}+\cdots$$ Now suppose we write the same function in spherical coordinates: $\hat f\equiv f\circ g$, where $g:R_\theta^3\to \mathbb{R}^3$ is the spherical coordinate mapping. Why can't we then write $$\hat f((\theta^1,\theta^2,\theta^3)+(\phi^1,\phi^2,\phi^3))=\hat f(\theta^1,\theta^2,\theta^3)+\phi_i\frac{\partial \hat f}{\partial \theta^i}+\phi_i\phi_j\frac{\partial^2 \hat f}{\partial \theta^i\,\partial \theta^j}+\cdots\tag{1}$$ What's going on here? Perhaps it's the periodicity, but then shouldn't the formula be valid if $(\phi^1,\phi^2,\phi^3)$ is sufficiently small?

What I'm really after is for a valid version of (1).


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