Stokes's Theorem 155Theorem. Define the vector field F — Pi + Qj + 0k and consider the
outside unit normal vector no to dG; since the unit tangent vector UQ
to dG exists at all but finitely many points of dG, so does no, being, by
definition, a unit vector perpendicular to UG- Then, by Gauss's Theorem,
i F • nG ds = //divFdA. (3.2.10)
JdG J J
G
EXERCISE 3.2.6.c Derive (3.2.10) from Green's Theorem. Hint: ifr(t) =
x(t) i + y(t) j is the parametrization of dG, then r' = x' t + y' j is tangent to dG;
the vector y' i — x'j is therefore perpendicular to r' and points outside (verify
this). Then F • na ds = -Qdx + Pdy. Note that div J^1 = Px + Qv.
3.2.3 Stokes's Theorem
The following result is known as Stokes's Theorem, after the English
mathematician GEORGE GABRIEL STOKES (1819-1903).
Theorem 3.2.3 Let S be a piece-wise smooth orientable surface whose
boundary OS consists of finitely many simple closed piece-wise smooth
curves so that the orientations of S and dS agree (see page 135). Let
F be a vector field, continuously differentiable in a domain containing S.
Then
I F • dr = 17 curlF • dtr; (3.2.11)
JdS J J
s
if the boundary of S consists of several closed curves, then the integral on
the left is the sum of the corresponding integrals over all those curves.
According to some accounts, the first statement of this theorem ap-
peared in a letter to Stokes, written in 1850 by another English scientist,
Sir WILLIAM THOMSON (1824-1907), also known as LORD KELVIN.
EXERCISE 3.2.7. (a)c Verify that Stokes's Theorem implies Green's theo-
rem. (b)B Prove Stokes's Theorem (the arguments are similar to the proof
of Greens Theorem).
The following result should be familiar from a course in multi-variable
calculus; Stokes's Theorem provides a key component for the proof.
Theorem 3.2.4 Let F be a continuous vector field in a simply connected
domain G, and assume that, for every unit vector n, the scalar field F • ri