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

How do we parameterize surfaces?

Today was like a review of the lesson we learned back in September/October. Parameterizations are given and you have to find how the x, y, and z vector-valued functions, which are in terms of u and v, relate in order to draw the surface. Doesn't this sound like finding traces and level curves?

The concept with gridlines was also introduced. First, the region that the surface covers is drawn. Then horizontal (y= __) and vertical lines (x= __) are drawn within the region (like a grid) and the equation of the level curve is found after plugging in the horizontal and vertical line values. After putting together several level curves, the curve can be seen and sketched.

Here's a cool site I found that kind of explains how artists apply surface parameterization to create their art: http://www.thegnomonworkshop.com/tutorials/paramaterization/parameterization.html.

Review: Green's Theorem

Today, we formed tripods to work on a worksheet on Green's Theorem. The problems on it were straightforward with basic applications of the theorem, but for question 2a, I got a negative answer. Does anyone know why? Or got a different answer? Just a reminder, Green's Theorem is:

Let R be a simply-connected region (no holes; doesn't cross itself) with a piecewise smooth (differentiable everywhere) boundary C (continuous, with a finite number of cusps) with positive orientation (counter clockwise orientation). If P and Q have continuous partial derivatives, then

How do we apply Green's Theorem?

Today we started the lesson by evaluating a line integral. As Mr. Honner reminded us, this is a crucial skill that we should remember, even though Green's theorem makes it unattractive in solving. We then brought back the idea of conservative vector fields. What happens when F is conservative? Quite simply, the integral equals zero. We can actually use dN/dx - dM/dy (written as dM/dx - dL/dy in yesterday's blog), to measure how "un-conservative" a vector field is. What happens if dN/dx - dM/dy is equal to 1? This actually equals the area of that region! Typical F(x,y) you may want to use (for simplicity) include <0,x> and <-y,0>.

If you are lost, visit IIT's courseware at http://www.youtube.com/watch?v=1aS7nTIYMx0 or MIT's courseware at http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/part-c-greens-theorem/

Both are great at explaining not only Green's theorem, but also typical (and not so typical) applications of it outside of the classroom.

Also from now on, a little homework section will be included, in order to let you know what the night's homework was. If you were absent, this would be a great way to briefly catch up with what was going on in class and get the homework.

Homework: p1099 # 7, 14, 20, 26, 32, 42

How do we turn line integrals into region integrals?

Today we went over line integrals and a new method of evaluating them. Usually, we find the integral and break it down into a piecewise curve. However, if we think of this curve as a region, we can then use a double integral to find it out. Green's theorem can be defined as follows:

Let R be a simply-connected region with a piecewise smooth boundary C, with positive orientation. If L and M have continuous partial derivatives, then 

• simply-connected meaning region has no holes in it and doesn't cross itself
• piecewise smooth meaning the region is continuous with a finite number of cusps
• positive orientation meaning counter clockwise

This provides us with a much easier method to solve some problems. Go try it out on some old homework! An efficient method of turning 40 minutes of long winded problems into 5 minutes of integration!