To evaluate the integral ( I(m) ) directly, we start by recognizing its geometric interpretation. The integral is given by:

[ I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y \frac{m\cos x \cos y - \cos x - \cos y}{\left( \sin^2 x + \sin^2 y + (m - \cos x - \cos y)^2 \right)^{3/2}} ]

Step 1: Identify the Integral as a Flux

The integrand can be interpreted as the ...

Yes. You can evaluate it by standard multivariable calculus, using only

• the identity behind your solid-angle formula,
• differentiation under the integral sign,
• integration by parts (periodicity),
• a local Taylor expansion near potential singular points, and
• a simple polar-coordinate integral.

Step 1: Put the integrand in Jacobian form Let n(x,y;m) = (sin x, sin y, m − cos x − cos y), R(x...

Evaluating the Chern Number Integral Directly

This integral represents the Chern number of a vector bundle over a torus. While the differential geometry approach gives us a clean result, we can evaluate it directly using multivariable calculus techniques.

Understanding the Integral Geometrically

The key insight is to recognize that the integral represents a solid angle (up to a factor of 1...

A straightforward way to obtain the value of

    1

I(m)= ───── ∬{-π}^{π} ∬{-π}^{π} 4π m cosx cosy – cosx – cosy ------------------------------------------------ dx dy (1) [ sin²x+sin²y+( m – cosx – cosy )² ]^{3/2}

is to look at it as an ordinary flux–integral and to keep track of the places where the denominator can vanish.
Everything ...

<think> I need to evaluate this integral directly without using differential geometry. The integral is:

[ I(m) = \frac{1}{4\pi} \int_{-\pi}^{\pi} dx \int_{-\pi}^{\pi} dy \frac{m \cos x \cos y - \cos x - \cos y}{\left( \sin^2 x + \sin^2 y + (m - \cos x - \cos y)^2 \right)^{3/2}} ]

And I know from theory that it should be:

[ I(m) = \begin{cases} 1 & \text{if } -2 < m < 0 \ -1 & \text{if } 0 ...

This is a fantastic question. The integral is indeed quite challenging to tackle with brute-force methods. The key, as you've hinted, is to use a combination of techniques from multivariable calculus that are inspired by the geometric nature of the problem, without necessarily invoking the full machinery of characteristic classes.

Here is a step-by-step evaluation that avoids the language of diff...

Put

 d(x , y ; m)= ( sin x ,   sin y ,   m−cos x−cos y ), 
 |d|= √[sin²x+sin²y+(m−cos x−cos y)²]

and observe that

 I(m)= 1/(4π) ∫_{T²} Ω(m;x,y)   with   Ω = d ·(∂_x d×∂_y d)/|d|³. (1)

Nothing deeper than ordinary multivariable calculus will be used; the vector notation only shortens the formulas.

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1. I(m) is p...

This is indeed a challenging integral that arises in the calculation of Chern numbers. Let me outline an approach to evaluate it directly using analytical techniques.

Direct Evaluation Approach

Step 1: Analyze the singularities

The integrand has singularities when the denominator vanishes:

This occurs only when:

• $\sin x = 0 \Rightarr...

One can in fact “cook up’’ an entirely elementary proof that does not invoke Chern classes, but only Stokes’ theorem (or Green’s theorem) on the two‐torus together with a little local (polar‐coordinate) analysis of the four points where the denominator can vanish. I will sketch the main steps; all that remains is routine but somewhat tedious differentiation and polar‐coordinate estimates.

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