alt="cosine alpha equals ModifyingAbove n With ampersand c period circ semicolon dot ModifyingAbove z With ampersand c period circ semicolon equals StartStartFraction left-parenthesis StartFraction a Over c EndFraction right-parenthesis OverOver StartRoot 1 plus left-parenthesis StartFraction a Over b EndFraction right-parenthesis squared plus left-parenthesis StartFraction a Over c EndFraction right-parenthesis squared EndRoot EndEndFraction"/>
α in (1.79) is the angle between the xy plane and surface 4. Substituting (1.79) into (1.78) gives
(1.80)
Then, from (1.69)
In (1.81), the first three terms are zero, hence (1.81) reduces to
or
From (1.76)
(1.83)
Hence, (1.82) can be written as
(1.84) leads to
(1.85)
The left‐hand side of the equation now can be found as
(1.86)
1.3.6 Stokes' Theorem
Stokes' theorem states that the line integral of a vector around a closed contour is equal to the surface integral of the curl of a vector over an open surface. It is expressed by
This can be illustrated in Figure 1.18. The small interior contours have adjacent sides that are in opposite directions yielding no net line integral contribution, as shown in
