Surface Integral: Transform, Calculate and Add Up Results?

In summary, surface integrals are integrated with respect to dS, which is then transformed into an integral with respect to dxdy. When evaluating the surface integral, it is important to project the surface onto a simpler surface, such as the x-y-plane. The surface element on the simpler surface is represented by dA, and the surface element on the original surface is represented by da, which is related by da = dA \sqrt{\frac{\partial f(x,y)}{\partial x}^2 + \frac{\partial f(x,y)}{\partial y}^2 +1}. Therefore, we only need to evaluate one integral in the x-y-plane. This can be generalized to any surface in terms of two parameters,
  • #1
coverband
171
1
As you know surface integrals are integrated with respect to dS. We then tranform the integral into one in dxdy. Is this the end of the problem or must we calculate it for dxdz and dydz as well and if so do you add up all results at the end!?
 
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  • #2
You can do it several ways. If you have some arbitrary surface, the trick is to project the surface to some simpler surface, for example the x-y-plane. With projection we have simpler integral in which we use dA, dA being infinitesimal surface element on our simpler surface, for example on the x-y-plane the area element is dx*dy (could be [tex]r dr d\phi[/tex] if we used polar cordinates).

You asked wether we calculate it for dxdz or dydz, the answer is: you have to use the plane on which the surface is projected on. If we have some surface f(x,y), we project it on the x-y-plane and this is almost always the case. So we have to evaluate only one integral, in this case one with dxdy.
 
  • #3
Just to clarify...

Thanks for reply


Just to clarify, if asked to evaluate
doub_int[f(x,y) dS]

We just solve in xy plane?

Thanks again...
 
  • #4
Yes. Solve in x-y-plane. But the most important thing to remember is the projection! Let da be surface element on f(x,y) and dA a surface element on x-y-plane. Then we have a relation [tex]da = dA \sqrt{\frac{\partial f(x,y)}{\partial x}^2 + \frac{\partial f(x,y)}{\partial y}^2 +1}[/tex] (follows from the cosine of the angle between the surface normal and the x-y-plane normal). So when you're doing the surface integral you get [tex]\int f(x,y)da = \int f(x,y) \sqrt{\frac{\partial f(x,y)}{\partial x}^2 + \frac{\partial f(x,y)}{\partial y}^2 +1} dA[/tex]. For only the surface area you have similar formula, you just have [tex]\int da[/tex] and so on.
 
  • #5
More generally, one can have a surface in terms of any 2 parameters. If x= x(u,v), y= y(u,v), z= z(u,v), then we can write the "position vector" of any point on the surface as
[itex]x(u,v)\vec{i}+ y(u,v)\vec{j}+ z(u,v)\vec{k}[/itex].

The two derivatives [itex]\vec{r}_u= x_u\vec{i}+ y_u\vec{j}+ z_u\vec{k}[/itex] and [itex]\vec{r}_v= x_v\vec{i}+ y_v\vec{j}+ z_v\vec{k}[/itex] lie in the tangent plane and their lengths are the differentials of length in that direction. Their cross product, [itex]\vec{r}_u\times\vec{r}_v[/itex] is called the "fundamental vector product" and its length, times dudv, is the differential of surface area.

In particular, if z= f(x,y), this gives exactly what JukkaVayrynen said.
 

1. What is a surface integral?

A surface integral is a mathematical tool used to calculate the flux, or flow, of a vector field across a surface. It involves transforming the surface into a parametric form, calculating the dot product of the vector field with the surface's normal vector, and then integrating over the surface.

2. How is a surface integral different from a regular integral?

A regular integral is used to find the area under a curve in a two-dimensional plane. A surface integral, on the other hand, is used to find the flux through a surface in a three-dimensional space. It involves integrating a vector field over a surface instead of a function over an interval.

3. What is the purpose of transforming the surface in a surface integral?

Transforming the surface into a parametric form allows us to express the surface as a function of two variables, making it easier to calculate the dot product with the vector field. It also allows us to change the limits of integration to match the new parametric form.

4. How do you calculate a surface integral?

To calculate a surface integral, you first need to transform the surface into a parametric form. Then, you calculate the dot product of the vector field with the surface's normal vector. Finally, you integrate this dot product over the surface using the appropriate limits of integration.

5. What are some real-life applications of surface integrals?

Surface integrals have various applications in fields such as physics, engineering, and computer graphics. They are used to calculate the electric or magnetic flux through a surface, the mass or pressure distribution on an object, and the amount of light reflected or transmitted through a surface. They are also used in fluid dynamics to calculate the flow rate through a given surface.

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