By Donald D. Fitts
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3-61) yields y (0V20 + V0*V0)riu - f (s0V0)*ds which is known as the first form o f Green’s theorem. (3-72) STOKES’ THEOREM 49 FIGURE 3-6 A second form o f Green’s theorem may be obtained as follows: First we exchange the roles of 0 and 0 in Eq. (3-72) to obtain / F ( 0 7 20 + 7 0 - 7 0 ) dv = fs (0 7 0 )-d s (3-73) We next subtract Eq. (3-73) from Eq. (3-72) and thereby obtain the desired result: f v (0 7 20 ~ 0 7 20) dv = fs (0 7 0 - 0 7 0 ) • ds Another useful integral relationship may be obtained by setting 0 = 1 in Eq.
W 6 is more complicated for this choice o f axes than it was for the original choice. We begin by considering Wi and w2: Wj = ioy + joy + kotz w2 = i/Jjp + jPy + kfiz Since atom B lies in the xz plane, the bond vector w t also lies in that plane and we have ay = 0. Since atoms A and C lie on the z = CYCLOHEXANE-CHAIR FORM 25 oig - -fiz . Furthermore, the magnitude of Wi and w 2 both have a value a: Wj • Wi = a *2 + a w2 • w2 y2 + a 2= = px 2+py2 + pz2-a 2 Thus we obtain the relations ag - a 2 -ax 2 Py2 =a2 - fig2 - px 2 =a2 - a z2 - p x 2 =%2- ^ 2 so that at this point Wj and w 2 have only two unknowns, ax and wt - iax + k(a 2 - otJC2)1/2 w2 = ipx + j(<*jc2 - Px2?
Thus, the final bond vectors are -:h /l Wi =1 W2 . y/2 a +k3 . a * ' 3 “ +) 3 J ' k 3 . y / 2X. V 6 . a W3 - * 3 '’ *> 3 ' + k 3 . a w4 = - i - r — a - k — (2-25) . y / 2 . V6 Ws=-iT fl-J1rfl+kJ W6 = 1^ 3- j ^ a - k ^ This representation of the bond vectors in the chair conformation of cyclohexane is completely equivalent to the representation in Eqs. (2-24). The two sets differ only in a rotation of the arbitrarily chosen set of cartesian coordinate axes. We now consider the hydrogen atoms in the chair conformation.