Line e Surface Infinitesimal

In summary, the conversation discusses the definition of line and surface infinitesimal, and how to calculate them using matrices and cross product. It also addresses the relationship between dxdy and d²xy.
  • #1
Jhenrique
685
4
I think you know definition of line infinitesimal:
[tex][ds]^2 = \begin{bmatrix} dx & dy & dz \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix}^2 \begin{bmatrix} dx\\ dy\\ dz\\ \end{bmatrix} = \begin{bmatrix} dr & d\theta & dz \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0 & r & 0\\ 0 & 0 & 1\\ \end{bmatrix}^2 \begin{bmatrix} dr\\ d\theta\\ dz\\ \end{bmatrix} = \begin{bmatrix} d\rho & d\phi & d\theta \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0 & \rho & 0\\ 0 & 0 & \rho\;sin(\phi)\\ \end{bmatrix}^2 \begin{bmatrix} d\rho\\ d\phi\\ d\theta\\ \end{bmatrix}[/tex]

From this, is correct if I deduce the formula to surface infinitesimal like this?
[tex][d^2S]^2 = \begin{bmatrix} dydz & dxdz & dxdy \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix}^2 \begin{bmatrix} dydz\\ dxdz\\ dxdy\\ \end{bmatrix} = \begin{bmatrix} d\theta dz & drdz & drd\theta \end{bmatrix} \begin{bmatrix} r & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & r\\ \end{bmatrix}^2 \begin{bmatrix} d\theta dz\\ drdz\\ drd\theta\\ \end{bmatrix} = \begin{bmatrix} d\phi d\theta & d\rho d\theta & d\rho d\phi \end{bmatrix} \begin{bmatrix} \rho^2\;sin(\phi) & 0 & 0\\ 0 & \rho\;sin(\phi) & 0\\ 0 & 0 & \rho\\ \end{bmatrix}^2 \begin{bmatrix} d\phi d\theta\\ d\rho d\theta\\ d\rho d\phi\\ \end{bmatrix}[/tex]

And more one second question: dxdy is equal d²xy ?
 
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  • #2
you can always try the "old way" of doing things...
Find the infinitesimal tangent vectors on your surface, and take the cross product
 

1. What is the concept of "Line e Surface Infinitesimal"?

"Line e Surface Infinitesimal" is a mathematical concept that refers to the study of infinitesimally small lines and surfaces. In other words, it deals with objects that are infinitely small and cannot be measured using traditional methods.

2. How is "Line e Surface Infinitesimal" different from traditional mathematics?

"Line e Surface Infinitesimal" is a branch of mathematics that focuses on objects that are infinitely small, while traditional mathematics deals with objects of measurable size. This means that the principles and methods used in "Line e Surface Infinitesimal" are unique and cannot be applied in traditional mathematics.

3. What are some real-world applications of "Line e Surface Infinitesimal"?

"Line e Surface Infinitesimal" has many practical applications in fields such as physics, engineering, and computer graphics. For example, it is used to model the movement of particles in fluid dynamics, to analyze the stress and strain on structures in engineering, and to create realistic 3D graphics in computer animation.

4. What are the limitations of "Line e Surface Infinitesimal"?

One of the main limitations of "Line e Surface Infinitesimal" is that it deals with objects that are infinitely small, which can be difficult to conceptualize and manipulate. Additionally, it is a highly theoretical branch of mathematics and may not have direct practical applications in some fields.

5. How does "Line e Surface Infinitesimal" relate to other branches of mathematics?

"Line e Surface Infinitesimal" is closely related to other branches of mathematics, such as calculus and differential geometry. It builds upon the principles and techniques of these branches to study objects that are infinitely small. It also has connections to other fields, such as physics and engineering, where it is used to solve complex problems.

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