Answered: Without using the formula (Stokes),…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Without using the formula (Stokes), calculate _l=∂S∫F(x).T(x)dl, where F(x) =x₁x₂i + x₂x₃j + x₁x₃k, e S : x₁² + x₂² + x₃² = 16, x₃ ≥ 0 (that is, S is the northern hemisphere center sphere 0 and radius 4).

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Given:

In the formulas below E, S and l always denote a solid, a surface, and a line, respectively. While n (x) denotes the normal unitary exterior of S in x, and T (x) denotes the unitary tangent of l in x.(image below)

1. S: g(z) = 0, n(x) = +ale. E, I: 7(t), te [a, b], T(r) = , r = 7(t).
(0)
Vela)
2. [n(x).k|ds = dr,drz, n(z).j|ds = dr,dr3. n(1).i|ds = drzdr3, i = (1,0,0), j = (0,1,0), k =
(0,0, 1), .
3. T(1).idl = drı, T(1).jdl = dr2, T(r).kdl = dr3, donde dl = ||Y(t)||dt, .
4. (Gauss) / V.F(r)dr = F(1).n(r)ds, onde dr = dr,dr,dx3, V =
E
5. (Stokes) S(V x F(x)).n(r)ds = S F(1).T(1)dl, F(r) = Fi(1)i + Fa(r)j + F3(1)k.
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