Page 146 - Vector Analysis

142 CHAPTER 4. Vector Calculus

we must have τ m ¨ (T ˆ N) ˝ ψm ą 0 on Bm X ty2 = 0u. In other words,

τ m = |τ m|(T ˆ N) ˝ ψm on Bm X ty2 = 0u .

Finally, since

τm ¨ τm 2 B ψm ¨ B ψm = gmgm22 = gm11 = ˇ B ψm ˇ2
B yβ B yδ ˇ ˇ,
= ÿ gm nα nγ gmαβgmγδ
ˇ B y1 ˇ
α,β,γ,δ=1

we conclude that τm = ˇ B ψm ˇ ˆ N) ˝ ψm on ty2 = 0u; thus (4.19) is established. ˝
ˇ ˇ(T

ˇ B y1 ˇ

Remark 4.84. On B Σ, the vector T ˆ N is “tangent” to Σ and points away from Σ. In
other words, T ˆ N can be treated as the “outward-pointing” unit “normal” of B Σ which
makes the divergence theorem on surfaces more intuitive.

4.7 The Stokes Theorem

4.7.1 Measurements of the circulation - the curl operator

We consider the circulation or the speed of rotation of a vector field u about an axis in the
direction N. Let P be a plane passing thorough a point a and having normal N, and Cr be
a circle on the plane P centered at a with radius r. Pick the orientation of the unit tangent
vector T which is compatible with the unit normal N (see Figure 4.6 for reference).

N

TP
T ra T

T

Figure 4.6: the circulation about an axis in direction N

Since the instantaneous angular velocity of a vector field u along the circle Cr is measured

by u ¨ T, it is quite reasonable to measure the circulation of u along Cr by averaging the

r

angular velocity; that is, we consider the quantity

1 ¿ u ¨ T ds (4.20)
2πr Cr r