nabla acting on products

Let f, g be differentiableMathworldPlanetmathPlanetmath scalar fields and u, v differentiable vector fields in some domain of 3.  There are following formulae:

  • GradientMathworldPlanetmath of a product function

  • DivergenceMathworldPlanetmath of a scalar-multiplied vector

  • Curl of a scalar-multiplied vector

  • Divergence of a vector product

  • Curl of a vector product

  • Gradient of a scalar productMathworldPlanetmath
    or, using dyads,

  • Gradient of a vector product

  • Divergence of a dyad product

  • Curl of a dyad product


  1. 1.

    v means the operator  vxx+vyy+vzz.

  2. 2.

    The gradient of a vector w is defined as the dyad  w:=iwx+jwy+kwz.

  3. 3.

    The divergence and the curl of a dyad product are defined by the equation
    *(uv):=i*(uv)x+j*(uv)y+k*(uv)z,  where the asterisks are dots or crosses and the partial derivativesMathworldPlanetmath of the dyad product the expression  (uv)x=uxv+uvx  and so on.

Title nabla acting on products
Canonical name NablaActingOnProducts
Date of creation 2013-03-22 15:27:05
Last modified on 2013-03-22 15:27:05
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Topic
Classification msc 26B12
Classification msc 26B10
Related topic Nabla
Related topic NablaNabla
Defines gradient of vector
Defines divergence of dyad product
Defines curl of dyad product