How to quantify the idea that physical calculations of objects of close by geometry give same answer? In many times in quick physics calculations, involving the geometry of a physical body, there is an assumption to simplfy the problem by considering the sample problem over a simpler geometry. Examples:

*

*"spherical cow" for calculating aerodynamic situation


*" assume mountain look likes a solid cylinder" for calculating the pressure by weight of mountains


*Assume earth is spherical etc for calculating density of it
In the ideas of approximation as said above, I don't understand a priori how physicist think that two things which have similar-ish geometry (cone of mountain to solid cylinder for example) should have give the right answer.
Is there a general principle of quantifying the "topological stability" of physical answers? In sense that the answer to a certain physical problem doesn't change much to variations in space which it is considered under?
 A: Often when one makes such approximations, they're usually shown to be valid under some set of assumptions, and the resulting calculations are usually 'continuous with respect to errors in the approximation'. It's difficult to phrase a general summary, so let me just describe it with the example of calculating the mass of a mountain. We shall describe a mountain using its mass density $\rho$. This is a function $\rho:\Bbb{R}^3\to[0,\infty)$, and we shall assume at the very least that $\rho$ is a bounded measurable function, and that it has compact support (mountains are finite in size, and have bounded densities). Then, the mass of the mountain is by definition just
\begin{align}
M_{\rho}&=\int_{\Bbb{R}^3}\rho\,dV,
\end{align}
where $dV$ is the volume element (i.e $3$-dimensional Lebesgue measure). Note that the shape of the mountain is encoded in the function $\rho$ itself, because we assume that the density is positive everywhere that the mountain is present, and is $0$ elsewhere, i.e the set $\text{supp}(\rho)$ describes the region occupied by the mountain.
Now, $\rho$ is a function, meaning at each point of $\Bbb{R}^3$, it has to give me a certain element of $\Bbb{R}$. No one can give such exact results. So, we have to admit that the $\rho$ we use is indeed an approximation, and that there are some uncertainties here. Ok, so how do these uncertainties affect the mass? Well, I claim not by much, because the mass depends on density 'continuously' in the following sense. The mountain lies inside of some fixed compact set $K\subset\Bbb{R}^3$ (we can be very crude with the determination of this set). Now, for any pair of densities $\rho_1,\rho_2$ with support in $K$, we have:
\begin{align}
\left|M_{\rho_1}-M_{\rho_2}\right|&=\left|\int_{\Bbb{R}^3}\rho_1\,dV-\int_{\Bbb{R}^3}\rho_2\,dV\right|\\
&=\left|\int_{K}(\rho_1-\rho_2)\,dV\right|\\
&\leq \text{vol}(K)\cdot \sup_{x\in K}|\rho_1(x)-\rho_2(x)|.
\end{align}
In fact, in more technical terms, what I've shown above is that the (linear) mapping $\rho\mapsto \int_{\Bbb{R}^3}\rho\,dV=\int_K\rho\,dV$, from the space of bounded measurable functions (equipped with the supremum norm) into the real numbers is continuous (actually I've shown Lipschitz continuity, but for linear maps, these are equivalent). So, if the RHS is small, so is the LHS. Another way of saying this result is that if you start with a density $\rho$, and you are sure that the uncertainty $\delta\rho$ is a small function (small in the sense of $\sup\limits_{x\in K}|(\delta\rho)(x)|$ being a small number), then the difference in the mass calculation when using $\rho+\delta\rho$ as the density versus using $\rho$ as the density is also a small quantity. So, in non-technical terms, small uncertainties in density imply small changes to mass calculation.
Anyway, I've gone over one approach for describing continuity in these calculations, but there's also other aspects in which we can make approximations. For instance, assuming the mountain is a cylinder in that calculation is fine, because the purpose of the calculation was to get an order of magnitude estimate for the mass/resulting pressure; in such situations we don't care about exact answers. We just want to know roughly how large the quantities are (for that just some reasonable upper and lower bounds are sufficient).
Assuming cows are spherical falls under a similar category; for the scales we're interested in for those problems, the exact geometry doesn't matter (see also the answer by @Nickolas Alves about multipole expansions). Note however, that we're not always this cavalier. Imagine you're a big sports company trying to design equipment for your star athlete (where e.g. outcomes of races are decided by milliseconds). Just take a look at some documentaries for how much biometric data they gather in order to come up with tailor-made shoes, sportswear or whatever else they come up with. No one in their situation will make the assumption that athletes are spheres! The bottom line is the level of detail and accuracy you put into your model depends highly on your purpose for the calculation.
A: A general principle that we often tell students is to model
the system as simple as possible to try to capture the physics of interest.
(that is, Can one possibly eliminate a complication that obscures the physics of interest?)
If that is not satisfactory, then one needs a refinement of the model

*

*e.g. maybe a point source is not good enough and one needs a multipole expansion like @NickolasAlves suggests. Computationally speaking, maybe one needs to break the system into multiple parts and use superposition.

*e.g. a linear model of variation isn't good enough and we need higher order terms. (i.e., perturbation theory: consider a one-parameter family of...)

What is presented in classes is likely the result of (someone) having tried various models and deciding that a particular model is best for introducing a certain concept. (For example, maybe don't start with the inverse-square law of gravitation to introduce the gravitational field in terrestrial situations.)
Along these lines, one could ask "How much does the answer change if one were to include an additional feature?", followed up by "can you live without including that feature?". This sort of thing shows up in considering experimental uncertainties... some uncertainties are worth considering and some are not.
A: I wouldn't go as far as saying that there is a completely general recipe, but in some cases, such as your spherical Earth example, there is a slightly more precise way. Namely, a multipole expansion. I'll exemplify it in the case of electrostatics, so that I can quote formulae from books I have nearby. Notice that Newtonian gravity is extremely analogous to electrostatics, apart from some differences in constants and signs.
Within electrostatics, one is often tasked with the problem of solving for the scalar potential $\varphi$ due to a charge distribution $\rho(t, \mathbf{x})$. The equation that rules this relation is Poisson's equation,
$$\nabla^2 \varphi = - \frac{\rho}{\epsilon_0}.$$
Now, suppose the charge distribution is supported within a radius $R$ of a chosen origin of space. This is the case, for example, for the Earth. One can show that for $r > R$ the scalar potential will be given by
$$\varphi(\mathbf{x}) = \frac{1}{\epsilon_0} \sum_{l=0}^{+\infty} \sum_{m=-l}^{l} \frac{1}{2l+1} \frac{q_{lm}}{r^{l+1}} Y_{l}{}^{m}(\theta,\phi), \tag{1}$$
where $(r,\theta,\phi)$ are the physicist's convention for spherical coordinates, $Y_{l}{}^m$ are the spherical harmonics and $q_{lm}$ are known as the spherical multipole moments, given by
$$q_{lm} = \int \rho(\mathbf{x}') r^{\prime l} Y^*_l{}^{m}(\theta',\phi') \mathrm{d}^3{x'}. \tag{2}$$
Eqs. (1) and (2) correspond to Eqs. (2.86) and (2.87) in Wald's Advanced Classical Electromagnetism, which also derives them.
Notice that large values of $l$ decay faster when observed at large distances, and hence they correspond to smaller effects. They also correspond to higher spherical harmonics, which describe more "detailed" deviations from spherical symmetry. In this sense, one has a somewhat precise notion of how a charged body's shape can influence the electrostatic field sourced by it. The most relevant term is spherically symmetric (in a physicist's mind, that is because, from afar, the body seems to be just a point). Next you have a dipole contribution, and so on.
As I mentioned, I don't believe this is not fully general as to apply to any example of geometry approximations performed by physicists, but it might give some more intuition on how some of these approximations work. If this interests you, the Wikipedia page for Multipole Expansions could be an interesting read.
