The map \ gamma _ 2 is called a " reparametrisation " of \ gamma _ 1; and this makes an equivalence relation on the set of all C ^ k differentiable curves in X.
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A differentiable curve is said to be " regular " if its derivative never vanishes . ( In words, a regular curve never slows to a stop or backtracks on itself . ) Two C ^ k differentiable curves
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A differentiable curve is said to be " regular " if its derivative never vanishes . ( In words, a regular curve never slows to a stop or backtracks on itself . ) Two C ^ k differentiable curves
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Such a vector field serves to define a generalized ordinary differential equation on a manifold : a solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.
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It is easy to establish this result for polygonal lines, but the problem came in generalizing it to all kinds of badly behaved curves, which include nowhere differentiable curves, such as the Koch snowflake and other fractal curves, or even a Jordan curve of positive area constructed by.
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For any 0-form, ?, defined on some differentiable curve ? ?" "'R " "'n " ( here the integral of ? over the boundary of the ? is understood to be the evaluation of ? at the endpoints of ? ).
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