The circulation integral of the position vector around a closed loop equals twice the vector area of the enclosed surface. The line integral of the cross product of the position vector and differential path vector is equivalent to twice the vector area of the surface enclosed by the loop. This allows for the simplification of complex calculations into direct geometric measurements. The result can be verified by checking the ratio of the calculated integral to the vector area, which converges to 2 as the approximation improves, providing a crucial verification of the result through the agreement between direct integration and the use of the theorem.
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$\complement\cdots$Counselor
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The circulation integral of the position vector around any closed loop is not zero; it is always equal to twice the vector area of the surface enclosed by the loop. This can be expressed as: $\oint_{\Gamma} x \times d x=2 A$.
This problem demonstrates a powerful principle in vector calculus: a complex line integral can often be converted into a much simpler geometric calculation (the vector area) using a specialized form of Stokes' theorem. The agreement between the two methods - direct integration and the use of the theorem - provides a crucial verification of the result.
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