Use Stokes’ Theorem to evaluate ∫c F · dr. In each Case C is oriented counterclockwise as viewed from above.
10. F(x, y, z) = 2y i + xz j + (x + y) k, C is the curve of intersection of the plane z = y + 2 and the cylinder x2 + y2 = l
To evaluate: The value of by the use of Stokes’ theorem.
The field is and C is curve of intersection of plane and cylinder .
Write the expression for curl of .
Write the expression for the Stokes’ theorem.
S is the surface.
Consider surface S, is in upward orientation. Write the expression for surface integral of over surface S.
A is the area.
Find the value of by using equation (1).
The surface S is a region which is enclosed by plane C. The expression for surface S, in polar coordinate system is,
Hence the equation is in the form of .
The curve C is oriented in counter-clockwise, so orient the surface S in upwards.
Compare the equations and .
Find the value of by using equation (3)
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