Surface integral with vector integrand

In summary, the conversation discusses the concept of integrating a vector field over a surface and its relationship to flux. It is explained that if the integrand is a vector, the resulting integral will give a vector answer, which is analogous to flux but with a direction. The process of converting the surface integral into a double integral is also described.
  • #1
thojrie
3
0
If we integrate a vector field over a surface, [tex]\int_S \vec{F} \cdot \vec{dS}[/tex], we get the flux through that surface. What does it mean if the integrad were a vector instead, [tex]\int_S \vec{F} dS[/tex]? I can't picture the Riemann sum.
 
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  • #2
But the surface is a vector because its an orientated surface right (surface + direction) ?
I mean when you do your surface integrals, you don't just define the surface, you define direction of the normal vector of the differential elements of the surface surface

Supposing you have some surface dS
you parameterise it with two variables, say s and t
then we can trace out our surface with some function x(s,t)
and then we take the dot product of the surface normal with our vector field:

[tex]
\int_S {\mathbf v}\cdot \,d{\mathbf {S}} = \int_S ({\mathbf v}\cdot {\mathbf n})\,dS=\iint_T {\mathbf v}(\mathbf{x}(s, t))\cdot \left({\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right) ds\, dt.
[/tex]

The key part here is that
[tex]
d\mathbf{S} = \mathbf{n}dS
[/tex]

With regards to visualising it as a Riemann sum, this will need to be checked by some one with more mathematical knowledge, but treat the whole dot product
[tex] {\mathbf v}(\mathbf{x}(s, t))\cdot \left({\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right) [/tex]
As the Jacobian which maps your complicated surface integral of a vector field to a simple two dimensional integral which you can then use your standard Riemann sums on
 
Last edited:
  • #3
thrillhouse86 said:
But the surface is a vector because its an orientated surface right (surface + direction) ?
I mean when you do your surface integrals, you don't just define the surface, you define direction of the normal vector of the differential elements of the surface surface

Supposing you have some surface dS
you parameterise it with two variables, say s and t
then we can trace out our surface with some function x(s,t)
and then we take the dot product of the surface normal with our vector field:

[tex]
\int_S {\mathbf v}\cdot \,d{\mathbf {S}} = \int_S ({\mathbf v}\cdot {\mathbf n})\,dS=\iint_T {\mathbf v}(\mathbf{x}(s, t))\cdot \left({\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right) ds\, dt.
[/tex]

The key part here is that
[tex]
d\mathbf{S} = \mathbf{n}dS
[/tex]

With regards to visualising it as a Riemann sum, this will need to be checked by some one with more mathematical knowledge, but treat the whole dot product
[tex] {\mathbf v}(\mathbf{x}(s, t))\cdot \left({\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right) [/tex]
As the Jacobian which maps your complicated surface integral of a vector field to a simple two dimensional integral which you can then use your standard Riemann sums on

I mean at the end of the day following the above steps turns your surface integral into a double integral (wrt to your parameterisation variables) of the form:
[tex] \int_{S} \mathbf{v}(s,t) \cdot d\mathbf{S} [/tex]

Hi thrillhouse86, thanks for the detailed reply! I don't think I phrased my question well enough though.

Because the dot product of two vectors is a scalar, when we find the flux through a surface the integrand and answer are both scalars. If instead, the integrand were a vector, the integral becomes,

[tex]\int_S \vec{F} dS[/tex]

which will give a vector answer. I assume this is analogous to flux (but with a direction), however I'm having trouble breaking it down and sussing out exactly what it means.
 
  • #4
I don't think it has a specific meaning like flux, it will simply "sum up" all of the different vectors that the field creates on the surface.

If the field would represent something like "Surface current density" then it will give you the total/effective current on the surface.
 
  • #5
thojrie said:
Hi thrillhouse86, thanks for the detailed reply! I don't think I phrased my question well enough though.

Because the dot product of two vectors is a scalar, when we find the flux through a surface the integrand and answer are both scalars. If instead, the integrand were a vector, the integral becomes,

[tex]\int_S \vec{F} dS[/tex]

which will give a vector answer. I assume this is analogous to flux (but with a direction), however I'm having trouble breaking it down and sussing out exactly what it means.
Writing [itex]\vec{F}= f\vec{i}+ g\vec{j}+ h\vec{k}[/itex], that integral would be
[tex]\int_S f dS\vec{i}+ \int_S g dS\vec{j}+ \int h dS\vec{k}[/itex].
 
  • #6
Hey thanks guys. That helped.
 

1. What is a surface integral with vector integrand?

A surface integral with vector integrand is a type of mathematical operation that involves the integration of a vector field over a surface. It is used to calculate the total flux or flow of a vector field through a given surface.

2. How is a surface integral with vector integrand different from a regular surface integral?

Unlike a regular surface integral, which involves the integration of a scalar function over a surface, a surface integral with vector integrand involves the integration of a vector field over a surface. This means that the result of a surface integral with vector integrand will also be a vector, whereas the result of a regular surface integral will be a scalar.

3. What are the applications of surface integrals with vector integrand?

Surface integrals with vector integrand have various applications in physics, engineering, and other fields. They are commonly used to calculate electric and magnetic flux, fluid flow, and other physical quantities. They can also be used in vector calculus to solve problems involving surfaces and volumes.

4. How is a surface integral with vector integrand calculated?

The calculation of a surface integral with vector integrand involves the use of a double integral, where the inner integral is taken over the surface and the outer integral is taken over the vector field. The specific method of calculation depends on the type of surface and vector field involved, but it typically involves parametrization and the use of vector calculus techniques.

5. Are there any limitations or restrictions to using surface integrals with vector integrand?

Like any mathematical operation, there are limitations and restrictions to using surface integrals with vector integrand. These include the requirement for the surface to be well-defined and continuous, the vector field to be well-behaved, and the need to carefully choose the orientation of the surface and the direction of the vector field. These limitations must be considered when applying surface integrals with vector integrand in practical situations.

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