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
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
So why does the integral equal zero when F is conservative?
ReplyDeleteBecause when F is conservative, that means dM/dy is equal to dN/dx. So when you plug that into Green's Theorem which is the double integral over region R of (dN/dx - dM/dy)dA, it turns to zero because the partial derivatives equal each other. Tada!