Use Stokes's theorem to calculate the line and inegral over the curve C of F dotted with a dr where F is equal to angle bracket z comma x comma y close angle bracket and C is oriented clockwise and is the boundary of a triangle with vertices point comma 0 comma 1, the point 3 comma 0 comma negative 2, and the point comma 1 comma 2. By Stokes's theorem we're going to turn this into a surface integral of the function, the curl of F dotted with the parameterization of that surface.

So we need to find a pair of things. And that is a permeterization of the surface, and using that the curl of F. First thing we need to do or next thing we need to do is to find what the surface would be with these three points. Well, in fact, these points are contained in a plane. Three points always determine a plane. And so we can use the equation Ax plus By plus Cz equals D using these values of x, y, and z respectively to form a system of equations and find the equation of that plane. This plane ends up being the function x minus y plus z equals 1.

So I'm waving my hands a little bit here, but this is something you could go back earlier in the course and be able to do. So maybe you should go practice that, if you can't figure out what that is. All right so this means that we are going to use in our parameterzation that z is equal to 1 minus x plus y. So the peramiterzation of s is going to be r of u comma v equals angle bracket u comma v comma 1 minus u plus v.

Now, we also need to determine the restrictions on these. So imagine what this boundary looks like. The triangle in the xy plane is bounded by 0,0, the point 3, 0, and the 0,1.

0, 0 is one of the boundaries. But really, what's going to dictate the x boundaries on this is where the other side intersects. And that actually is a line. Well, actually the x's are going to range from 0 to 3. That's just given.

So we have 0 is less than or equal to u is less than or equal to 3. But the y-values will be determined by the line that's formed there. So the other restriction will be 0 is less than or equal to v, is less than or equal to 1 minus 1/3 times u. 1 minus 1/3x is the equation that is the other side of that triangle.

Now from here, we can say that t sub u is that equal to angle bracket 1, 0, negative 1 with t sub v equal to angle bracket 0, 1, 1, which means that t sub u crossed with t sub v is equal to angle bracket 1, negative 1, 1, end angle bracket. That's necessary for our integral, of course.

Now also, we need to notice that the curl of F-- go back and be able to calculate that. But the curl of F with this parameterization is equal to angle bracket 1, 1, of 1 so that the integral of the curve c of F dotted with dr by Stokes's theorem is equal to the double integral over the surface s of curl of F dotted with the ds.

Well, that will be the integral from 0 to 3 with respect to u of the integral from 0 to 1 minus 1/3 u with respect to v of angle bracket 1, 1, 1 and angle bracket dotted with angle bracket 1, negative 1, comma 1, end angle bracket. And that dot product is actually 1. So the integral from 0 to 3 with respect to u of the integral from 0 to 1 minus 1/3 u with respect to v of the function 1, which is equal to the integral from 0 to 3 with respect to i of 1 minus 1/3 times u, which will be equal to 3 over 2.