Texts jo f dr in this case c is oriented counterclockwise as...

Texts: Jo F. dr. In this case, C is oriented counterclockwise as viewed from above. Use Stokes's Theorem to evaluate F(x, y, z) = yzi + (2 − 5y)j + (x² + y²)k, x² + y² ≤ 36 S: the first-octant portion of x² + z² = 36 over x² + y² = 36 Use Stokes's Theorem to evaluate dr In this case, C is oriented counterclockwise as viewed from above. F(x, y, z) = yzi + (2 − 5y)j + (x² + y²)k, x² + y² ≤ 36 S: the first-octant portion of x² + z² = 36 over x² + y² = 36

Transcript

00:01 So in this question, we have, we want an integral, which is the integral around c of f dotted with the line element dr.

00:16 Now, stokes ' theorem tells us that for c, a closed curve, this is the integral along the interior of c, a, where let's say that the double integral over a of the curl of f dotted with the normal, unit normal vector da.

00:44 So, and in this case, we have c is going to be the boundary of a.

00:50 So what we have is f is 2yi plus 3zj plus xk.

01:03 So let's take its curl, the curl of f, let's write it as column vectors, dx, dy, dz crossed with 2y, 3z, x.

01:17 So for the x component, we have dy x minus dz of 3z, so that's minus 3.

01:23 For the y component, we have dz of 2y minus dx of x, so that's minus 1.

01:31 And for the z component, we have dx of 3z minus dy of 2y, which is minus 2.

01:36 So the curl of f has components minus 3i minus j minus 2k.

01:53 Now we need to have a look at the area which c is the boundary of.

01:59 C is a triangle with vertices 2, 0, 0, 0, 2, 0, and 0, 0, 2.

02:15 So let's draw that in 3d space.

02:19 Z, x, y.

02:22 We've got all of these 2s, so our triangle c is in here.

02:35 And what we can see by inspection is that the normal vector of that is going to be pointing through the origin and out normally, because there is a symmetry if we rotate around the line, the line which is kind of intersecting the solid angle between the three axes.

03:04 And if that means that we can exchange any of the axes and we have a symmetry, then n must be symmetrical under the exchange of any axes, and that means that n must have the same component in each slot.

03:20 So symmetry under exchange means that n hat must be proportional to 1 to just i plus j plus k.

03:37 But since it's a normal vector, that means we have to divide by root 3, so n hat is 1 over root 3 times i plus j plus k...

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