End of the Year

We hope you did great on the exam. Our whole team thanks you for keeping up with our blog. If you have any comments on how we can improve this "classroom blog" for next year, please feel free to comment below!

Review: Divergence Theorem + Stokes' Theorem

Today we did some review on the Divergence Theorem and Stokes' Theorem. These problems intertwined the various concepts we have been learning, and learned to switch between formulas to get the most convenient equation to solve. Test on Friday!!!

How do we apply Stoke's Theorem?

So we learned the last theorem ever in this course today-- Stokes' Theorem. Just as magnificent as Mr. Honner made the theorem sound, it is actually pretty amazing. It basically incorporates both the Divergence Theorem and Green's Theorem into one. It states:

Let S be a smooth surface that is bounded by a simple closed curve. Then the closed integral of F dot dr is equal to the double integral of curl F dot dS, over the surface S.

In the Divergence Theorem, surface integrals and triple integrals are shown to be interrelated and it shows the flux. Green's Theorem relates line integrals (work) and double integrals. But in this theorem, it links the two theorems--surface integrals and line integrals, which makes it pretty cool.

Well, that was the last theorem you'll ever learn in this course. Make use of it on the final on Friday!

The homework for tonight was the same as yesterdays: P. 1130 # 2, 5, 8, 10, 25, 26.

How do we compute flux?

Today we went over evaluating SSs f(x,y) ds some more, and then learned another integral: the flux integral. The integral is SS F dot n ds. This eventually gives us the following, more practical equations:
1) SSd F dot (-Gxi-Gyk+k)dA
2) SSd F dot (Ru x Rv) dA

Using these integrals, you can evaluate the flux, or the flow, of the surface in the direction given.

How do we evaluate surface integrals?

We began today's lesson with a simple set up of an integral to find surface area. As we quickly learned, sometimes we can use polar to make bounds easier to define, or make an integral much easier to solve. We then moved on to surface integrals. Surface integrals are evaluated just like line integrals, with the exception of a double integral instead of a single one.

Here it is in single integral form:

Simply account for the second integral, and you are set to begin solving surface integrals!

How do we find areas of parametric surfaces?

Today's lesson was short, and was simply a recap of sketching surfaces through different methods. We can use grid lines, which are reminiscent of traces that we used previously, to piece it together piece by piece, or simply find the relationships between the x, y, and z components. We then practiced finding the surface area of a surface we sketched. Finally, we worked on some problems to further grasp our understanding of the subject matter.

One cool program you can use to help you see surfaces can be seen here:

How do we compute surface area of a parametric surface?

Note: This was Friday's (05/27/11) lesson

Today we went over some simple parametrizations, and how to prove a given vector parametrization is a specific shape. We then went over the idea of finding the surface area of a surface. This would simply be the small area in the parallelogram between the Rx and Ry components. The cross product of these components can also give us the normal vector, which would make it much easier to find the tangent plane to a surface. We ended on the note that the Area of a Parametric surface is:

Surface Area= SSsds = SS_D ||Ru x Rv|| dA

Where s is the traverse of the surfaces and ds is the addition of the little pieces of the surface