Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense ( analogous to interchanging the limits in a definite integral ):
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If you do a line integral around a closed path and you don't get zero, is the quantity that you get equal to the energy dissipated by, say, heat from friction?
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If the line integral is a closed curve in a region where the function is singularities, then the value of the integral is simply zero; this is a consequence of the Cauchy integral theorem.
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The mathematical statement of the law is a relation between the total amount of magnetic field around some path ( line integral ) due to the current which passes through that enclosed path ( surface integral ).
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(Note that, per Stokes's theorem, the closed line integral of H�dl around a contour is equal to the open surface integral of curl H�dA across the surface bounded by the closed contour.
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Some variants of \ oiint and \ oiiint have arrows on them to indicate the sense of integration, such as a line integral around a closed curve in the clockwise sense, and higher dimensional analogues.
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The converse does hold e . g . if the domain is simply connected; this is Cauchy's integral theorem, stating that the line integral of a holomorphic function along a closed curve is zero.
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If " V " is defined in terms of the line integral, the ambiguity of " V " reflects the freedom in the choice of the reference point \ mathbf r _ 0.
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The gradient theorem implies that line integrals through gradient fields are Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows.
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The fundamental theorem of calculus for line integrals requires path independence in order to express the values of a given vector field in terms of the partial derivatives of another function that is the multivariate analogue of the antiderivative.
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