Surface Integrals: Clearing Up Misunderstanding

[Bassalisk]

Hi,

I understand that from my EM class there exist a surface integral which is actually a way of summing infinitesimally small surface elements ds.

But then I ran into some theorems on internet and I saw the denotation of double integral, over a surface S. And they called that a surface integral. But I learned a surface integral to be a single integral.

Can somebody clear this misunderstanding for me? Mainly this came from Stokes' theorem. Theorem it self isn't a problem, just the denotation of surface integral as double integral is.

 

[Studiot]

Hello Bassalisk,

I understand that from my EM class there exist a surface integral which is actually a way of summing infinitesimally small surface elements ds.

Not quite.

The surface integral is not just the sum of small pieces of surface. It is rather more than that.

First let us take a physical quantity that can vary from place to place (ie point to point in space).
This quantity may be a scaler or a vector.
If we can write an expression for this quantity at every point in some region of space we can do calculus on the expression.
The expression is called a vector point function or a scalar point function.
If the region is one dimensional it is a curve or line, twisting and turning in 3D space.
Integrals in this region are called line integral.
If the region is two dimensional it is a surface in 3D space and integrals are called surface integrals
If the region is three dimensional it is called a volume integral.

Line integrals can be evaluated by a single integration
Surface integrals need two integration for evaluation
Volume integrals need three integrations for evaluation

Stokes, Gauss and Greens theorems connect these types of integral, allowing us to reduce the type of integral and thus the number of required integrations, in certain circumstances.

The circumstances are that for instance to reduce the surface integral to a line integral (by Stokes ) we need an expression for the boundary curve in 3D of the region (surface area) and we perform our line integration around that boundary. Stokes theorem relates the value of the surface integral over the area to the line integral around its boundary.

When you come to study finite element methods (FEA) in numerical analysis you will find this provides a powerful alternative method that can often greatly reduce calculation effort, known as the boundary element method (BEM).

go well

 

[Bassalisk]

Hello Bassalisk,
Not quite.

The surface integral is not just the sum of small pieces of surface. It is rather more than that.

First let us take a physical quantity that can vary from place to place (ie point to point in space).
This quantity may be a scaler or a vector.
If we can write an expression for this quantity at every point in some region of space we can do calculus on the expression.
The expression is called a vector point function or a scalar point function.
If the region is one dimensional it is a curve or line, twisting and turning in 3D space.
Integrals in this region are called line integral.
If the region is two dimensional it is a surface in 3D space and integrals are called surface integrals
If the region is three dimensional it is called a volume integral.

Line integrals can be evaluated by a single integration
Surface integrals need two integration for evaluation
Volume integrals need three integrations for evaluation

Stokes, Gauss and Greens theorems connect these types of integral, allowing us to reduce the type of integral and thus the number of required integrations, in certain circumstances.

The circumstances are that for instance to reduce the surface integral to a line integral (by Stokes ) we need an expression for the boundary curve in 3D of the region (surface area) and we perform our line integration around that boundary. Stokes theorem relates the value of the surface integral over the area to the line integral around its boundary.

When you come to study finite element methods (FEA) in numerical analysis you will find this provides a powerful alternative method that can often greatly reduce calculation effort, known as the boundary element method (BEM).

go well

Just like old times :)

You just cleared few misunderstandings I had before, that even weren't asked in this thread.

But why are some surface integrals written as single integral and some as double integral?

Specifically, take in the flux of the Electric field. It is

[itex]\oint \vec{A}\cdot d\vec{S}[/itex] (closed surface) and I found in other theorems that they use double integral and calling it a surface integral.

Is this denotation interchangeable?

 

[Studiot]

Is this denotation interchangeable?

First remember that in general in Engineering all the integrals that are used are definite integrals.
That is they are a number, which of course is what engineers want.

Now there is some variability about notation for integrals and some difficulty representing them in PF Tex as well.

Some (particularly older) texts use the single ∫ with the circle to mean line integrals, some use it to mean line integrals for a closed loop only - which is modern practice and makes sense.
Obviously any closed loop is a boundary defining a surface so can be connected to a surface integral over that surface.

Another convention is to write a subscript s or v to denote surface or volume integrals, though I have never seen an l used for line.

Whatever it is a good idea to make sure the number of integral signs match the number of differential signs ie one if you use ds and two if you use dydx or three if you use dydxdz.

What I have seen and consider bad practice are such uses as

[tex]\oint\limits_s {F.dxdy} [/tex]

or

[tex]\int {\int {F.ds} } [/tex]

 

[Bassalisk]

First remember that in general in Engineering all the integrals that are used are definite integrals.
That is they are a number, which of course is what engineers want.

Now there is some variability about notation for integrals and some difficulty representing them in PF Tex as well.

Some (particularly older) texts use the single ∫ with the circle to mean line integrals, some use it to mean line integrals for a closed loop only - which is modern practice and makes sense.
Obviously any closed loop is a boundary defining a surface so can be connected to a surface integral over that surface.

Another convention is to write a subscript s or v to denote surface or volume integrals, though I have never seen an l used for line.

Whatever it is a good idea to make sure the number of integral signs match the number of differential signs ie one if you use ds and two if you use dydx or three if you use dydxdz.

What I have seen and consider bad practice are such uses as

[tex]\oint\limits_s {F.dxdy} [/tex]

or

[tex]\int {\int {F.ds} } [/tex]

I think I understand. Thought that people would actually point this out. It is confusing to newcomers like me.

Thank you.

 

[Studiot]

Yes unfortunately sometimes you have to infer (work it out) from the context what type of integral is meant.

Line, surface and volume integrals differ from multiple integrals as follows

A multiple integral is of the form

∫∫(expression in x and y) dxdy

It may extend to infinity and may be an indefinite integral in whcih case the result will contain some arbitrary constants (2 in my example).

A line, surface or volume integral is of the form

∫_sF.n ds

∫∫(expression in x and y). (expression locating each point) dxdy

In my example you have an expression for the quantity that varies over the surface and an expression that identifies (every) point on the surface - the normal n in this case.

To evaluate the integral you have to provide two expressions one for F and one for n, in terms of x and y or R and θ or whatever.
An ordinary multiple integral only has one expression.

 

[Bassalisk]

Yes unfortunately sometimes you have to infer (work it out) from the context what type of integral is meant.

Line, surface and volume integrals differ from multiple integrals as follows

A multiple integral is of the form

∫∫(expression in x and y) dxdy

It may extend to infinity and may be an indefinite integral in whcih case the result will contain some arbitrary constants (2 in my example).

A line, surface or volume integral is of the form

∫_sF.n ds

∫∫(expression in x and y). (expression locating each point) dxdy

In my example you have an expression for the quantity that varies over the surface and an expression that identifies (every) point on the surface - the normal n in this case.

To evaluate the integral you have to provide two expressions one for F and one for n, in terms of x and y or R and θ or whatever.
An ordinary multiple integral only has one expression.

ooooh I see. The original title of the thread was: Are double integrals and surface integrals one and the same?

I guess this answers my question. Thank you for that. Was really confusing and all. I have been studying for 8 hours x 2 days. My brain is swelling.

Thank you very much, you helped a lot. :)

 

[Studiot]

There is just one more chapter in this story.

If you can find and expression for x and y etc in terms of a single variable - parameter - often t is used - you can again reduce a multiple integral to a single.
When you do this you not only transform the expression for x and y but also need to express dx and dy in terms of dt.

go well

 

[Bassalisk]

There is just one more chapter in this story.

If you can find and expression for x and y etc in terms of a single variable - parameter - often t is used - you can again reduce a multiple integral to a single.
When you do this you not only transform the expression for x and y but also need to express dx and dy in terms of dt.

go well

Yes I am familiar with that. Thank you.