Volume of a closed surface (divergence theorem)

In summary, the conversation discusses the possibility of a formula for calculating the volume of a closed surface, similar to Green's theorem for calculating the area of a closed curve. While there are surface integrals that can be used for this purpose, the math is more complex. The conversation also includes a link to a resource discussing line, surface, and volume integrals, as well as a proposed equation for calculating the volume.
  • #1
Jhenrique
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  • #2
Green's and Stokes' theorems can by used on surface integrals, just like they are used for 2-D integrals.

The math is a little more complex, however.

Google: 'line surface and volume integrals'
 
  • #5
I am searching for something similar to this:
f76df7ea16919c17fe62cef9eb303fd7.png



EDIT: I think that the volume can be calculated by:
[tex]V=\iint_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset xdydz = \iint_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset ydzdx = \iint_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset zdxdy = \frac{1}{3}\iint_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset (xdydz+ydzdx+zdxdy)[/tex]

Correct?
 
Last edited:
  • #6
Jhenrique said:
[tex]V=\iint_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset xdydz = \iint_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset ydzdx = \iint_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset zdxdy = \frac{1}{3}\iint_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset (xdydz+ydzdx+zdxdy)[/tex]

This equation is really correct?
 

What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a mathematical theorem that relates the surface integral of a vector field to the volume integral of the divergence of that field over a closed surface. It is a fundamental theorem in vector calculus and is often used in physics and engineering to solve problems involving fluid flow, electric and magnetic fields, and other vector fields.

What is a closed surface?

A closed surface is a three-dimensional surface that encloses a finite volume. It can be thought of as a boundary that completely surrounds a region of space. In the context of the divergence theorem, the closed surface is the surface over which the surface integral is calculated.

What is the relationship between the surface and volume integrals in the divergence theorem?

The divergence theorem states that the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of that field over the enclosed volume. In other words, it relates the behavior of a vector field on the surface to its behavior inside the enclosed volume.

What are some applications of the divergence theorem?

The divergence theorem has numerous applications in physics and engineering. It is commonly used to calculate fluid flow rates, electric and magnetic flux, and the behavior of other vector fields. It is also used in the study of fluid dynamics, electromagnetism, and other areas of physics and engineering.

How is the divergence theorem derived?

The divergence theorem can be derived using the fundamental theorem of calculus and the definition of the divergence of a vector field. One approach is to divide the enclosed volume into small cubes and use the mean value theorem to approximate the volume integral. Another approach is to use the divergence theorem in higher dimensions to prove it in three dimensions.

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