Use the method of Example 5 to calculate ∫C F · dr, where
and C is any positively oriented simple closed curve that encloses the origin.
To find: The line integral of vector field over curve C .
The given vector field is,
Consider a positively oriented curve C which is piece-wise smooth, simple closed curve in plane with domain D. Then,
is continuous first-order partial derivative of P,
is continuous first-order partial derivative of Q, and
P and Q have continuous partial derivatives.
Write the expression for circle with center origin.
r is radius.
Refer to Figure 11 in the textbook of section 16.4 for region D.
Consider a curve , which is a circle with counter-clockwise orientation centered at origin and radius a. The radius a is small and lies inside the circle C. The region between the circles C and is D as shown in Figure 11.
The curve C is positively oriented, piecewise-smooth, and simply closed curve with domain D and hence Green’s theorem is applicable.
Compare the two expressions and .
Find the value of .
Take common term outside in the numerator.
Find the value of .
Substitute for and for in equation (1),
Find the value of
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