Evaluating integral on surfaces

In summary, the integral \int_{B_{\delta}(p_0)}{\frac{1}{(1 + d^2(p_0, p))^2}\mathrm{d}v_g(p)} can be evaluated using the divergence theorem on a compact surface (M, g) with a small enough \delta, which yields a value of \pi.
  • #1
Goklayeh
17
0
Hello everyone! Could someone tell me how to evaluate the integral
[tex]
\int_{B_{\delta}(p_0)}{\frac{1}{(1 + d^2(p_0, p))^2}\mathrm{d}v_g(p)}
[/tex]
where [itex]B_{\delta}(p_0) \subset M[/itex] and [itex](M, g)[/itex] is a generic compact surface and [itex]\delta > 0[/itex] can be made small as one wishes (as usual, [itex]d[/itex] is the intrinsic metric)? Thanks you in advance

P.S. I strongly suspect that the value should be [itex]\pi[/itex], but I'm a newbie in integration-on-manifold without any idea about how to proceed.
 
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  • #2
The integral can be evaluated using the divergence theorem. Since the integrand is a function, the divergence theorem states that \int_{B_{\delta}(p_0)}{\frac{1}{(1 + d^2(p_0, p))^2}\mathrm{d}v_g(p)} = \int_{\partial B_{\delta}(p_0)}{\frac{1}{(1 + d^2(p_0, p))^2}\mathrm{d}s_g(p)}where \partial B_{\delta}(p_0) is the boundary of the ball and \mathrm{d}s_g(p) is the area element on the boundary. Now, since the boundary of the ball is a circle, the integral can be evaluated using the formula\int_{\partial B_{\delta}(p_0)}{\frac{1}{(1 + d^2(p_0, p))^2}\mathrm{d}s_g(p)} = \int_0^{2\pi}{\frac{1}{(1 + d^2(p_0, p))^2}\delta\,\mathrm{d}\theta}where \theta is the angle around the circle. The integral can then be evaluated explicitly:\int_0^{2\pi}{\frac{1}{(1 + d^2(p_0, p))^2}\delta\,\mathrm{d}\theta} = \pi\deltaTherefore, the value of the integral is \pi.
 

Related to Evaluating integral on surfaces

1. What is the purpose of evaluating integrals on surfaces?

The purpose of evaluating integrals on surfaces is to calculate the total value of a function over a given surface. This allows for the determination of important quantities such as surface area, volume, and mass.

2. What are the different methods for evaluating integrals on surfaces?

There are several methods for evaluating integrals on surfaces, including the use of parametric equations, double integrals, and surface integrals. Each method has its own advantages and is used depending on the specific problem at hand.

3. How is a surface integral different from a regular integral?

A surface integral is a type of integral that is evaluated over a surface, rather than a one-dimensional interval as in a regular integral. This means that instead of integrating along a single axis, the function is integrated over a two-dimensional region.

4. What are some applications of evaluating integrals on surfaces in real-world situations?

Evaluating integrals on surfaces has many practical applications, such as calculating the surface area of a three-dimensional object, finding the center of mass, and determining the amount of fluid flow through a surface. It is also used in fields such as physics, engineering, and computer graphics.

5. What are some common challenges when evaluating integrals on surfaces?

Some common challenges when evaluating integrals on surfaces include choosing the appropriate method for the given problem, setting up the integral correctly, and dealing with complex surfaces or functions. It is important to have a strong understanding of mathematical concepts and techniques to overcome these challenges.

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