Connection between sums and integrals over closed paths and their areas

Question

Is there a deeper connection between the following identies involving sums or integrals over a closed path (resp. circle) resp. the area enclosed – e.g. a general principle that is underlying all of them?

1. Euler's totient is the Fourier transform of the greatest common divisor function:

$$\sum_{k=1}^n \operatorname{gcd}(k,n)\cdot e^{i2\pi k/n} = \varphi(n)$$

2. The sum of the $n$-th roots of unity is $0$:

$$\sum_{k=1}^n e^{i2\pi k/n} = 0$$

3. Cauchy's integral theorem:

$$\oint_\gamma f(z)dz = 0 $$

4. Cauchy's residue theorem:

$$\oint_{\gamma} f(z)dz = 2\pi i \sum_{k=1}^n\operatorname{I}(\gamma,a_k)\operatorname{Res}(f,a_k)$$

5. The Gauss-Bonnet theorem:

$$\int_{\partial M}k_g\ ds = \int_M K\ dA + 2\pi\chi(M)$$

For $K\equiv 0$:

$$\int_{\partial M}k_g\ ds = 2\pi\chi(M)$$

6. The Chern–Gauss–Bonnet theorem:

$$\int _{M}{\mbox{Pf}}(\Omega )=(2\pi )^{n}\chi (M)$$

Or is there possibly no deep connection between (all of) them, and the analogies are superficial?

Furthermore: Is this selection too arbitrary, and examples of such identities abound, actually?

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A connection between 2, 3, 4 is demonstrated here.

Answer 1

You might look at the Stokes' theorem : https://en.wikipedia.org/wiki/Stokes%27_theorem . It links the integral on the boundary of a compact $\partial K$ of a differential form $\alpha$ to the integral of its differential $d\alpha$ on the compact. It is a kind of generalisation of the fact that the integral of a function on a segment is the difference at the boundary of the segment of a primitive. It is an essential formula which is behind almost all formula with integrals. (Green-Riemann, Green-Ostrogradski, integration by parts, Bochner's formula, etc.).