Volume integral turned in to surface + line integral?

In summary, the conversation discusses the representation of a surface element and a line element in a volume integral, specifically in the context of magnetic helicity in a plasma flux rope. The book being referenced is "Lectures in Magnetohydrodynamics" by D.Schnack, and there is a question about the notation used in the book. The possibility of representing the volume element as a line element in the direction of the field is also mentioned.
  • #1
AntiElephant
25
0
Hi, I have a book that makes the equality.

[itex] \vec{B}dV = (\vec{e_1}B_1 + \vec{e_2}B_2 + \vec{e_1}B_2)dx_1 dx_2 dx_3 \\[1ex]
= dx_1 \vec{e}_1(B_1 dx_2 dx_3 ) + dx_2 \vec{e}_2(B_2 dx_1 dx_3 ) + dx_3 \vec{e}_3 (B_3 dx_1 dx_2) = (\vec{B}\cdot d\vec{S}) d\vec{l}. [/itex]

I'm a bit confused as to how it makes that last equality. In a very general sense, the surface element is given by;

[itex] d\vec{S} = (dx_2dx_3,dx_1dx_3,dx_1dx_2) [/itex]

right? What I need is a way of represententing [itex] d\vec{l} = (dx_1,dx_2,dx_3)[/itex] as being multiplied component-wise by the 3 summation terms of [itex] \vec{B} \cdot d\vec{S} [/itex], but as far as I can tell the notation [itex] (\vec{B}\cdot d\vec{S})d\vec{l} [/itex] doesn't seem to do that?

If this is not possible, it might be cause it's specific to my situation. I'm looking at the integral of [itex] B [/itex] over the volume a plasma flux rope - which is defined as the volume encompassed by a fixed selection of magnetic field lines.
 
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  • #2
I think its just a shorthand that they are using. I can't seem to use the formal rules of vectors to transform the right side into the left. What book is this from?
 
  • #3
xiavatar said:
I think its just a shorthand that they are using. I can't seem to use the formal rules of vectors to transform the right side into the left. What book is this from?

Lectures in magnetohydrodynamics by D.Schnack. Think it's possible to access it online. Chapter 12 when it talks about magnetic helicity (page 73).

If it's not possible in general to write it like that, it might because it's relevant to the situation.

Maybe the volume element is chosen to be a line element [itex] dl [/itex] in the direction of the field, mutiplying a surface element [itex] dS [/itex] which is perpendicular to the line and field direction (covering the poloial cross section). In this coordinate system [itex] B_2 = B_3 = 0 [/itex] always. And

[itex] \vec{B} dV = dx_1 \vec{e_1} (B_1 dx_2 dx_3) = (\vec{B} \cdot d\vec{S})\cdot d\vec{l} [/itex]

I'm not entirely sure how correct this is.
 
Last edited:

1. What is the difference between a volume integral and a surface + line integral?

Volume integrals calculate the total value of a function over a three-dimensional region, while surface + line integrals calculate the value of a function over a two-dimensional surface or a one-dimensional curve. Essentially, volume integrals look at the entire volume of a solid, while surface + line integrals focus on specific surfaces or curves within that volume.

2. How is a volume integral turned into a surface + line integral?

In order to turn a volume integral into a surface + line integral, we use the Divergence Theorem or Stokes' Theorem. These theorems relate a volume integral to a surface integral or a line integral by taking the integral of the divergence or curl of a vector field over the volume. This allows us to convert a three-dimensional integral into a two-dimensional or one-dimensional integral.

3. What are the applications of converting a volume integral into a surface + line integral?

This conversion is often used in physics and engineering to solve problems involving fluid flow, electric and magnetic fields, and other vector fields. It also allows us to express physical laws and principles in terms of surface or line integrals, which can make calculations and interpretations easier.

4. Can a volume integral be turned into a surface + line integral for any three-dimensional region?

While the Divergence Theorem and Stokes' Theorem allow us to convert most volume integrals into surface + line integrals, there are some regions where this may not be possible. For example, if the region has holes or disjointed parts, the theorems may not apply. In these cases, alternative methods may need to be used.

5. Are there any limitations or special cases to consider when converting a volume integral into a surface + line integral?

One limitation to consider is the orientation of the surface or curve being integrated over. The orientation must be consistent with the orientation used in the volume integral. Additionally, the vector field used must be well-defined and continuous over the entire region. In some cases, the theorems may also require the use of a closed surface or curve. These limitations and special cases should be carefully considered when converting a volume integral into a surface + line integral.

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