View source for Hamilton operator

''nabla operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462402.png" />-operator, Hamiltonian''

A symbolic first-order differential operator, used for the notation of one of the principal differential operations of vector analysis. In a rectangular Cartesian coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462403.png" /> with unit vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462404.png" />, the Hamilton operator has the form

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The application of the Hamilton operator to a scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462406.png" />, which is understood as multiplication of the  "vector"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462407.png" /> by the scalar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462408.png" />, yields the [[Gradient|gradient]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462409.png" />:

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i.e. the vector with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624011.png" />.

The scalar product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624012.png" /> with a field vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624013.png" /> yields the [[Divergence|divergence]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624014.png" />:

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The vector product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624016.png" /> with the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624018.png" />, yields the [[Curl|curl]] (rotation, abbreviated by rot) of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624019.png" />, i.e. the vector

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624021.png" />,

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The scalar square of the Hamilton operator yields the [[Laplace operator|Laplace operator]]:

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The following relations are valid:

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The Hamilton operator was introduced by W. Hamilton [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.R. Hamilton,   "Lectures on quaternions" , Dublin  (1853)</TD></TR></table>



====Comments====
See also [[Vector calculus|Vector calculus]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.E. Rutherford,   "Vector mechanics" , Oliver &amp; Boyd  (1949)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Apostol,   "Calculus" , '''1–2''' , Blaisdell  (1964)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Holman,   H. Rummler,   "Alternierende Differentialformen" , B.I. Wissenschaftsverlag Mannheim  (1972)</TD></TR></table>