{"id":5075,"date":"2014-03-27T06:05:50","date_gmt":"2014-03-26T21:05:50","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5075"},"modified":"2014-08-01T18:40:12","modified_gmt":"2014-08-01T09:40:12","slug":"formulas-involving","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5075","title":{"rendered":"Formulas involving ∇"},"content":{"rendered":"
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If the partial derivatives of \\bold{A}, \\bold{B}, U and V are assumed to exist, then<\/p>\n

    \n
  1. \\displaystyle \\nabla(U + V) = \\nabla U + \\nabla V or grad(U + V) = grad U + grad V<\/li>\n
  2. \\displaystyle \\nabla\\cdot(\\bold{A} + \\bold{B}) = \\nabla\\cdot\\bold{A} + \\nabla\\cdot\\bold{B} or div(\\bold{A} + \\bold{B}) = div\\bold{A} + div\\bold{B}<\/li>\n
  3. \\displaystyle \\nabla\\times (\\bold{A} + \\bold{B}) = \\nabla\\times\\bold{A} + \\nabla\\times\\bold{B} or curl(\\bold{A} + \\bold{B}) = curl\\bold{A} + curl\\bold{B}<\/li>\n
  4. \\displaystyle \\nabla\\cdot(U\\bold{A}) = (\\nabla U)\\cdot\\bold{A} + U(\\nabla\\cdot\\bold{A})<\/li>\n
  5. \\displaystyle \\nabla\\times(U\\bold{A}) = (\\nabla U)\\times\\bold{A} + U(\\nabla\\times\\bold{A})<\/li>\n
  6. \\displaystyle \\nabla\\cdot(\\bold{A}\\times\\bold{B}) = \\bold{B}\\cdot(\\nabla\\times\\bold{A}) - \\bold{A}\\cdot(\\nabla\\times\\bold{B})<\/li>\n
  7. \\displaystyle \\nabla\\times(\\bold{A}\\times\\bold{B}) = (\\bold{B}\\cdot\\nabla)\\bold{A} - \\bold{B}(\\nabla\\cdot\\bold{A}) - (\\bold{A}\\cdot\\nabla)\\bold{B} + \\bold{A}(\\nabla\\cdot\\bold{B})<\/li>\n
  8. \\displaystyle \\nabla(\\bold{A}\\cdot\\bold{B}) = (\\bold{B}\\cdot\\nabla)\\bold{A} + (\\bold{A}\\cdot\\nabla)\\bold{B} + \\bold{B}\\times(\\nabla\\times\\bold{A}) + \\bold{A}\\times(\\nabla\\times\\bold{B})<\/li>\n
  9. \\displaystyle \\nabla\\cdot(\\nabla U) \\equiv \\nabla^2U \\equiv \\frac{\\partial^2U}{\\partial x^2} + \\frac{\\partial^2U}{\\partial y^2} + \\frac{\\partial^2U}{\\partial z^2} is called Laplacian<\/em> of U and
    \n\\displaystyle \\nabla^2 \\equiv \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2} + \\frac{\\partial^2}{\\partial z^2} is called Laplacian operator<\/em>. <\/li>\n
  10. \\displaystyle \\nabla\\times(\\nabla U) = 0. The curl of the gradient of U is zero. <\/li>\n
  11. \\displaystyle \\nabla\\cdot(\\nabla\\times\\bold{A}) = 0. The divergence of the curl of \\bold{A} is zero. <\/li>\n
  12. \\displaystyle \\nabla \\times (\\nabla \\times \\bold{A}) = \\nabla (\\nabla \\cdot \\bold{A}) - \\nabla^2\\bold{A}<\/li>\n<\/ol>\n<\/blockquote>\n