Calculate the work done by the force field F(x,y,z)=(y2,z2,x2) along the curve of intersection of the sphere x2+y2+z2=1, the cylinder x2+y2=x, and the halfspace z>0. The path is traversed in a direction that appears clockwise when viewed from the high above the xy-plane.
How do I even find the path for starters... after that it is straight forward.
I know what the sphere would look like in the xyz plane but not the cylinder. How do you sketch the cylinder?
I've now done this and got the same result as TeeEm. Here's a sketch with many details omitted:
1. The path is a tear drop shaped curve, symmetrical in +ve and -ve y. The path segments for +ve, -ve y are, in vector form:
r+=(x,x(1−x),1−x) r−=(x,−x(1−x),1−x)
where x∈[0,1]
2. I parameterise this with x=sin2θ with θ∈[0,π/2] which is a bijection. This gives:
3. I'll leave the rest to you; you have to split up the path, and integrate ∫F⋅dr over both bits with the appropriate limits to take you clockwise around it when viewed from above. Some nice cancellation occurs.
3. I'll leave the rest to you; you have to split up the path, and integrate ∫F⋅dr over both bits with the appropriate limits to take you clockwise around it when viewed from above. Some nice cancellation occurs.
3. I'll leave the rest to you; you have to split up the path, and integrate ∫F⋅dr over both bits with the appropriate limits to take you clockwise around it when viewed from above. Some nice cancellation occurs.
3. I'll leave the rest to you; you have to split up the path, and integrate ∫F⋅dr over both bits with the appropriate limits to take you clockwise around it when viewed from above. Some nice cancellation occurs.
what exactly bothers you because I do not follow the issue...
Ok so the intersection of the two is given as that pic which u have and I copied down just now. We're finding the path of when it is (in the zx plane) (0,1) to (1,0) then (1,0) to (0,-1) right?
But we are given the condition that z>0 so that means everything in the bottom quadrants should be ignored but then why are we finding the path of (1,0) to (0,-1)?
Ok so the intersection of the two is given as that pic which u have and I copied down just now. We're finding the path of when it is (in the zx plane) (0,1) to (1,0) then (1,0) to (0,-1) right?
But we are given the condition that z>0 so that means everything in the bottom quadrants should be ignored but then why are we finding the path of (1,0) to (0,-1)?