Exploring Surface Terms: Understanding the Meaning Behind the Term 'Surface Term

[Ratzinger]

what is meant by the term 'surface term'?

thank you

 

[HallsofIvy]

I have no idea! Could you give us some context- the complete sentence or even paragraph where you saw it?

 

[Ratzinger]

Well, I remember reading it in a quantum field textbook (Zee?), but don't know more where exactly. The first two google pages give things like "a Ginzburg-Landau model with a surface term" or "Chern-Simons Integral as a surface term". So it seems to be physics language.

I thought it is a household (mathematical) term, because I remember the author used it without further explaining.

Have to note that my math and physics skills are not too advanced, can deal with upper undergraduate physics texts, but have hard times with more. (Especially when they throw in terms and equations from out of nowhere.)

 

[lurflurf]

Usually a surface term is a term that results from integration over the boundry of some region. Them a bulk or volume term results from integrating over the region. Surface and boundry terms are related by stokes theorem and by conservation laws. A common example is a surface term equal to the flux of some quantity through the boundry of a surface equals the rate of change in the amount of the quantity inside the surface. It might help to think of the 1-D situation
$$\frac{d}{dt}\int_{a(t)}^{b(t)} f(x,t) dt=\int_{a(t)}^{b(t)} \frac{\partial}{\partial t}f(x,t)dx+f(x,t)\frac{\partial x}{\partial t}\displaystyle{|_{x=a(t)}^{x=b(t)}}$$
So in 1-D integrals are bulk terms and differences are surface terms. Often in applications the bulk terms are integrals over 3-D regions and surface terms are integrals over 2-D boundries of 3-D regions.

 

[Ratzinger]

thanks lurflurf...why aren't you science adviser or homework helper yet?

 

[PBRMEASAP]

I've seen it used in the context of integration by parts:

∫_a^b u dv = uv|_a^b - ∫_a^b v du

in which case the first term on the right is called the "surface term".

 

[EnumaElish]

This is off the subject, but -- PBRMEASAP, how did you make the math notation without tex?

 

[inha]

right click on the eq and view source.

 

[EnumaElish]

Does that mean I need to type character string &-#-8-7-4-7 (without the dashes) to make the ∫ sign?

 

[inha]

hmm. I painted and viewed the selection source again but it seems like he copy/pasted the integral sign from somewhere else. the rest is html. oh well, tex is better looking and easier anyway.

 

[PBRMEASAP]

& int ; [ sub ] a [ /sub ] [ sup ] b [ /sup ]

without the spaces gives ∫_a^b.

I haven't figured out how to make the superscript go directly over the subscript. You can also make greek letters:

& alpha ;
& beta ;
& gamma ;

gives

α, β, γ, etc.

 

[hypermorphism]

You can't really do that without an appropriate stylesheet, since they are separate characters in a standard font. MathML has appropriate styles for this, but is not exactly commonly used. The complete list of standard HTML 4 entities can be found here. A complete list of all possible Unicode hex characters (entered by &#x followed by the hex code) can be found at Unicode's charts site. Note that while most fonts implement some subsets of the Unicode character set, not all characters are encoded in the most common fonts (so the less common characters are likely to be rendered as the empty box on others' browsers).