WORKSHEET XVI
1. Find the curl of the vector field F = (x2-y) i + 4z j + x2 k.
2. State Stokes' Theorem.
3. Using Stokes' Theorem find the curl integral for the hemisphere S: x2 + y2 + z2 = 16, z>0, and the vector field F = y i - x j.
4. For each of the following, use the surface (curl) integral in Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction:
(A) F = x2 i + 4x j + z2 k, C is the ellipse 4x2 + y2 = 4 in the xy-plane, counterclockwise when viewed from above.
(B) F = 2y i + 3x j - z2 k, C is the circle x2 + y2 = 9 in the xy-plane, counterclockwise when viewed from above.
(C) F = (y2 + z2) i + (x2 + y2) j + (x2 + y2) k, C is the square bounded by x=+1 and y=+1 in the xy-plane, counterclockwise when viewed from above.
George Gabriel Stokes (1819 - 1903)
Stokes, one of the most influential scientific figures of his century, was Lucasian professor of mathematics at Cambridge University from 1849 until his death in 1903. His theoretical and experimental investigations covered hydrodynamics, elasticity, light, gravity, sound, heat, meteorology, and solar physics. He left electricity and magnetism to his friend Lord Kelvin. It is another one of those delightful quirks of history that the theorem that we call Stokes’ theorem isn’t his theorem at all. He learned of it from William (Lord Kelvin) Thomson in 1850 and a few years later included it among questions on an examination. It has been known as Stokes’ theorem ever since. As usual, things balanced out. Stokes was the original discover of the principles of spectrum analysis that we now credit to Robert Bunsen and Gustav Robert Kirchhoff.