00:01
So in this section, we've been using stokes theorem as a really helpful tool to simplify the integrals that we're dealing with, specifically surface integrals.
00:13
So stokes theorem tells us this right here, where c is the boundary of a surface -oriented counterclockwise, and we have the freedom to choose the surface whose boundary is c.
00:26
So we have s being the region inside the rectangle formed by the given points.
00:32
We are given f to be the vector z squared 2xy 4y squared so that means that the curl of f will be equal to the partial derivative of y with respect or the partial derivative of r with respect to y so that will be 8y minus the partial derivative of q with respect to z and then looking at the curl formula we get that this will be 2 z minus 0 and then 2y minus 0 so the curl of f is equal to the vector 8y 2 z 2 y and then we want to find the equation of the plane passing through the four given points so we let the equation of the plane we know to be a x plus b y plus c z equals d.
01:45
And the point one zero zero on the plane indicates that a equals d.
01:52
The point one to one indicates that a plus two b plus c equals d.
01:58
And then the point zero to one indicates that two b plus c equals d.
02:04
So if we subtract equation a from equation two, we see that a is equal to zero, and that implies that d is also equal to zero.
02:18
So just by doing that simple, but yet clever algebraic manipulation, we can change our equation of the plane to be much simpler.
02:29
So it'll be by plus cz equal zero because both a and b are equal to zero.
02:36
We can use equation 3 to replace c with the negative 2b.
02:44
So we'll get minus 2b z equal 0.
02:49
So that tells us that y is going to be equal to, or z is going to be equal to y over 2.
03:01
That's because we can move this over and then divide by 2b.
03:06
Divide by 2b...