6.2 Green’s Theorem Suppose F : R^{2} → R^{2}, F (x, y) = (M(x, y), N(x, y)). S ⊂ R^{2} is a nice compact surface. ∫_{∂S} F · ds = ∫_{∂S}(M, N) · (dx, dy) = ∫_{∂S} Mdx + Ndy = ∫ ∫_{S}[\frac{∂N}{∂x} − \frac{∂M}{∂y}]dA = ∫ ∫_{S}[∇ × F] · kdA

7.3 Stoke’s Theorem Suppose F : R^{3} → R^{3}, F (x, y, z) = (M(x, y, z), N(x, y, z), P (x, y, z)). S ⊂ R^{3} is a bounded, piecewise smooth, oriented surface. ∫_{∂S} F · ds = ∫ ∫_{S}[∇ × F] · dS

= ∫ ∫_{X}[(∇ × F)(X(s, t))] · N(s, t)dsdt Ex: Suppose ∂S = ∅ ∫ ∫_{S}[∇ × F] · dS = ∫_{∂S} F · ds =

Ex: Suppose ∂S = {(x, y, 0) | x^{2} + y^{2} = 1} = ∂S̃ ∫ ∫_{S}[∇ × F] · dS = ∫_{∂S} F · ds = ∫ ∫S̃[∇ × F] · dS̃ Suppose S̃ = {(x, y, 0) | x^{2} + y^{2} ≤ 1} A parametrization for S̃ X(s, t) = (s(cost), s(sint), 0) T_{s}(s, t) = (cost, sint, 0) T_{t}(s, t) = (−s(sint), s(cost), 0) N(s, t) = (0, 0, s) Suppose F (x, y, z) = (y^{2}, 3, z^{2})

3 Another parametrization for S̃ Y (s, t) = (s, t, 0) T_{s}(s, t) = (1, 0, 0) T_{t}(s, t) = (0, 1, 0) N(s, t) = (0, 0, 1) Suppose F (x, y, z) = (y^{2}, 3, z^{2}) ∇ × F = (0, 0, −2y) ∫ ∫S̃[∇ × F] · dS̃ = ∫ ∫S̃(0, 0, −2y) · dS̃ = ∫ ∫_{Y} [(∇ × F)(X(s, t))] · N(s, t)dsdt = ∫_{−1}^{1} ∫_{−\sqrt{1−t^{2}}}^{√\overline{1−t^{2}}}(0, 0, −2t) · (0, 0, 1)dsdt = − ∫_{−1}^{1} ∫_{−\sqrt{1−t^{2}}}^{√\overline{1−t^{2}}} 2tdsdt = − ∫_{−1}^{1} 2ts|_{−\sqrt{1−t^{2}}}^{√\overline{1−t^{2}}} dt = − ∫_{−1}^{1} 4t\sqrt{1 − t^{2}} dt Let u = 1 − t^{2}, du = −2tdt, t = ±1 implies u = 0 = − ∫_{0}^{0} −2\sqrt{u} du = 0 6.2 Divergence Theorem in the Plane Suppose F : R^{2} → R^{2}, F (x, y) = (M(x, y), N(x, y)). S ⊂ R^{2} is a nice compact surface. n ⊂ R^{2}, outward unit normal to ∂S. ∫_{∂S} F · nds = ∫ ∫_{S}[∇ · F]dA

7.3 Gauss’s Theorem Suppose F : R^{3} → R^{3}, F (x, y, z) = (M(x, y, z), N(x, y, z), P (x, y, z)). D ⊂ R^{3} is a bounded, solid nice 3-dimensional region. ∫ ∫_{∂D} F · dS = ∫ ∫ ∫_{D}[∇ · F]dV Calculate ∫ ∫_{S} F · dS where

F (x, y, z) = (xy^{2}, y^{3}, 4x^{2}z), S = S_{1} ∪ S_{2} ∪ S_{3} and

S_{1} = {(x, y, 5) | x^{2} + y^{2} ≤ 4}

S_{2} = {(x, y, z) | x^{2} + y^{2} = 4, 0 ≤ z ≤ 5}

S_{3} = {(x, y, 0) | x^{2} + y^{2} ≤ 4} Let S_{a} = sphere of radius a centered at the point P = (P_{1}, P_{2}, P_{3}) Prop 3.4. Divergence of F at P = (∇ · F)(P_{1}, P_{2}, P_{3}) = lim\frac{3}{4πa^{3}} ∫ ∫_{S_{a}} F · dS

= lim_{a→0^{+}} ^{\frac{∫ ∫_{Sa} F ·dS}{\frac{4πa^{3}}{3}}}

= lim_{a→0^{+}} \frac{flux}{volume of ball of radius a} = flux density Let C_{a} = the circle of radius a centered at the point P = (P_{1}, P_{2}, P_{3}) lying in the plane perpendicular to the unit vector n and containing the point P.

Prop 3.5. The component of the curl of F at P in the direction of n =

n · (∇ × F)(P) = lim_{a→0^{+}} \frac{1}{πa^{2}} ∫_{C_{a}} F · ds

= lim_{a→0^{+}} \frac{1}{πa^{2}} ∫_{X}(F · T)ds

= \frac{circulation of F along C_{a}}{area of surface bounded by C_{a}}