Differential 1-Form

• Every differential equation determines a direction field in the region — the line attached at the point has slope . This is called the direction field of .

  Every integral curve of the direction field satisfies

• (Arnold 1.1) For any vector tangent to the graph of a smooth function the ratio is equal to the derivative of the function at the corresponding point.

  Here means multiplying with the component of corresponding to .

• A differential 1-form is a function where attached vectors are linear at each given point where they are attached. That is, they can be described using smooth functions and are of the form

  • They can be integrated along oriented closed segments of curves. Let be a segment of the curve and be a smooth mapping from the interval . The integral of the form over is defined as

    Where

  • (Arnold 1.2) Parameter / Coordinate Independence of Differential 1-form: The Integral of a 1-form over an oriented closed segment of a curve is independent of the parameter for parameters giving the same orientation. When the orientation is reversed, the integral changes sign.

  • (Arnold 1.3) The integral of a 1-form over a segment of a curve on which can be taken as parameter coincides with the usual definite integral of the function

• The differential manifold is another way to look at generalizing line integrals for manifolds for which the dot product may not be defined.

  The key insight to generalizing line integrals to such manifolds is to ”absorb the dot product into the integrand”. What we get is the definition of the 1-form.

Links §

• Arnold - Ch. 1