Representation of vector fields

  • Authors

    • Alexander G. Ramm Mathematics Department, Kansas State University, CW 207, Manhattan, KS 66506-2602, USA
    2015-05-01
    https://doi.org/10.14419/gjma.v3i2.4577
  • Vector elds, Representation of vector elds.
  • A simple proof is given for the explicit formula which allows one to recover a \(C^2\) – smooth vector field \(A=A(x)\) in \(\mathbb{R}^3\), decaying at infinity, from the knowledge of its \(\nabla \times A\) and \(\nabla \cdot A\). The representation of \(A\) as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded \(C^2\) - smooth domain.

  • References

    1. [1] O. Ladyzhenskaya, The mathematical theory of viscous incompressible fluid, Gordon and Breach, New York, 1969.

      [2] D. Menzel, Fundamental formulas of physics, Prentice Hall, New York, 1955.

      [3] R. Temam, Navier-Stokes equations, North Holland, Amsterdam, 1984.

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  • How to Cite

    Ramm, A. G. (2015). Representation of vector fields. Global Journal of Mathematical Analysis, 3(2), 73-76. https://doi.org/10.14419/gjma.v3i2.4577