If we divide S into small patches Sij , then Sij is nearly planar. It is called the surface integral or flux integral of F over S. If S is given by a vector function r u, v , then n is given by Equation 6. The figure shows the vector field F in Example 4 at points on the unit sphere. Since S is a closed surface, we use the convention of positive outward orientation. The disk S2 is oriented downward.
Although we motivated the surface integral of a vector field using the example of fluid flow, this concept also arises in other physical situations. For instance, if E is an electric field Example 5 in Section Thus, if the vector field F in Example 4 represents an electric field, we can conclude that the charge enclosed by S is:.
Then, the rate of heat flow across the surface S in the body is given by the surface integral. The temperature u in a metal ball is proportional to the square of the distance from the center of the ball.
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Surface Integrals In this section, we will learn about: Integration of different types of surfaces. Documents Similar To 02Surface Integral.
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Heiz Kierk. This means that we have a closed surface. This is easy enough to do however. First define,. This means that we will need to use. Also, the dropping of the minus sign is not a typo. When we compute the magnitude we are going to square each of the components and so the minus sign will drop out. Here are polar coordinates for this region. We could have done it any order, however in this way we are at least working with one of them as we are used to working with.
We can now do the surface integral on the disk cap on the paraboloid. This one is actually fairly easy to do and in fact we can use the definition of the surface integral directly.
It also points in the correct direction for us to use. Because we have the vector field and the normal vector we can plug directly into the definition of the surface integral to get,. To get the square root well need to acknowledge that. Here is the value of the surface integral. Finally, to finish this off we just need to add the two parts up. Here is the surface integral that we were actually asked to compute.
So, as with the previous problem we have a closed surface and since we are also told that the surface has a positive orientation all the unit normal vectors must point away from the enclosed region. To help us visualize this here is a sketch of the surface. In this case since the surface is a sphere we will need to use the parametric representation of the surface.
This is,. Here are the two individual vectors and the cross product. Therefore, we will need to use the following vector for the unit normal vector. Again, we will drop the magnitude once we get to actually doing the integral since it will just cancel in the integral.
Now, we need to do the integral over the bottom of the hemisphere. This also means that we can use the definition of the surface integral here with. The last step is to then add the two pieces up.
Here is surface integral that we were asked to look at. We will leave this section with a quick interpretation of a surface integral over a vector field. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.
Show Solution So, as with the previous problem we have a closed surface and since we are also told that the surface has a positive orientation all the unit normal vectors must point away from the enclosed region.
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