c Dr Oksana Shatalov, Spring 2013 1 Spring 2013 Math 251 Week in Review 12 courtesy: Oksana Shatalov (covering Sections 14.8 (continued), 14.9 ) 14.8: Stokes’ Theorem (continued) Key Points • Let S be an oriented piece-wise-smooth surface that is bounded by a simple, closed, piecewise smooth boundary curve C with positive orientation. Let F be a vector field whose components have continuous partial derivatives on an open region in R3 that contains S. Then I ZZ F · dr = curlF · dS. C S Examples I 1. Evaluate I = C F · dr if F = h2y + 3ex , z − y 8 , x + ln(z 2 + 1)i and C is the curve of inter- section of the plane x+y+z = 0 and sphere x2 +y 2 +z 2 = 1. (Orient C to be counterclockwise when viewed from above). c Dr Oksana Shatalov, Spring 2013 2 2. Verify Stokes’ Theorem for the surface S: x2 + y 2 + 5z = 1, z ≥ −5 (oriented by upward normal) and the vector field F~ = xz~i + yz~j + (x2 + y 2 )~k. c Dr Oksana Shatalov, Spring 2013 3 RR R ~= 3. Verify Stokes’ Theorem S curlF~ · dS F~ · d~r for the vector field F~ = hyz 2 , −xz 2 , z 3 i ∂S and the cylinder x2 + y 2 = 9 for 1 ≤ z ≤ 2 oriented out. c Dr Oksana Shatalov, Spring 2013 4 14.9: The Divergence Theorem Key Points • The Divergence Theorem: Let E be a simple solid region whose boundary surface S has positive (outward) orientation. Let F be a continuous vector field on an open region that contains E. Then ZZ ZZZ F · dS = divF dV. S E RR ~ where F~ = x3~i + y 3~j + z 3~k and S is 4. Use the Divergence Theorem to compute S F~ · dS, the surface of the region enclosed by x2 + y 2 = 1 and the planes z = 0, z = 2. c Dr Oksana Shatalov, Spring 2013 5 5. (cf. problem 4, WIR #11) Use the Divergence Theorem to find flux of the vector field F = hx, y, 1i across the surface S which is the boundary of the region enclosed by the cylinder y 2 + z 2 = 1 and the planes x = 0 and x + y = 5. 6. Verify the Divergence Theorem for the region E = {(x, y, z) : 0 ≤ z ≤ 9 − x2 − y 2 } and the vector field F~ = ~r = x~i + y~j + z~k 7. Apply the Divergence Theorem to compute RR S F · dS for the vector field F(x, y, z) = hx3 + sin(yz), y 3 , y + z 3 i over the complete boundary S of the solid hemisphere {(x, y, z) : x2 + y 2 + z 2 ≤ 1, z ≥ 0} with outward normal.