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Vector Fields

 
 

Divergence

#~v _ = _ #~v ( ~x, ~y, ~z ) _ is a vector field over &reals.&powthree., _ #~v _ = _ ~v_~x #~i + ~v_~y #~j + ~v_~z #~k , _ define the #~{divergence} of #~v:

div #~v _ _ = _ &nabla. &dot. #~v _ _ = _ fract{∂~v_~x,∂~x} _ + _ fract{∂~v_~x,∂~y} _ + _ fract{∂~v_~x,∂~z}

Curl

#~v _ = _ #~v ( ~x, ~y, ~z ) _ is a vector field over &reals.&powthree., _ #~v _ = _ ~v_~x #~i + ~v_~y #~j + ~v_~z #~k , _ define the #~{curl} of #~v:

curl #~v _ _ = _ rndb{fract{∂~v_~z,∂~y} - fract{∂~v_~y,∂~z}} #~i _ + _ rndb{fract{∂~v_~x,∂~z} - fract{∂~v_~z,∂~x}} _ #~j _ + _ rndb{fract{∂~v_~y,∂~x} - fract{∂~v_~x,∂~y}} #~k

This can be written:

curl #~v _ _ = _ &nabla. # #~v _ _ = _ det{ _ #~i, _ #~j, _ #~k / fract{∂,∂~x}, fract{∂,∂~y}, fract{∂,∂~z} / ~v_~x, ~v_~y, ~v_~z}

Irrotational Fields

A vector field #~v whose curl is everywhere zero, i.e. _ curl #~v _ == _ 0 , _ is said to be #~{irrotational}

In particular for any scalar field &phi.

curl grad &phi. _ _ = _ &nabla. # ( &nabla.&phi. ) _ _ = _ det{ _ #~i, _ #~j, _ #~k / fract{∂,∂~x}, fract{∂,∂~y}, fract{∂,∂~z} / ~v_~x, ~v_~y, ~v_~z}

_

_ _ = _ rndb{fract{∂&powtwo.&phi.,∂~y∂~z} - fract{∂&powtwo.&phi.,∂~z∂~y}} _ #~i _ + _ rndb{fract{∂&powtwo.&phi.,∂~z∂~x} - fract{∂&powtwo.&phi.,∂~x∂~z}} _ #~j _ + _ rndb{fract{∂&powtwo.&phi.,∂~x∂~y} - fract{∂&powtwo.&phi.,∂~y∂~x}} _ #~k _ == _ #0

Conversely, if the vector field #~v is irrotational, then &exist. a scalar field &psi. (not unique) such that _ ~#v _ = _ grad &psi..

Solenoidal Fields

A vector field #~v whose divergence is everywhere zero, i.e. _ div #~v _ == _ 0 , _ is said to be #~{solenoidal}

In particular for any vector field ~#w

div curl ~#w _ _ _ = _ _ _ &nabla. &dot. ( &nabla. # ~#w ) _ _ _ = _ _ _ rndb{#~i fract{∂,∂~x} + #~j fract{∂,∂~y} + #~k fract{∂,∂~z}} &dot. det{ _ #~i, _ #~j, _ #~k / fract{∂,∂~x}, fract{∂,∂~y}, fract{∂,∂~z} / ~w_~x, ~w_~y, ~w_~z}

_

_ _ = _ rndb{fract{∂&powtwo.~w_~z,∂~x∂~y} - fract{∂&powtwo.~w_~y,∂~x∂~z}} _ + _ rndb{fract{∂&powtwo.~w_~x,∂~y∂~z} - fract{∂&powtwo.~w_~z,∂~y∂~x}} _ + _ rndb{fract{∂&powtwo.~w_~y,∂~z∂~x} - fract{∂&powtwo.~w_~x,∂~z∂~y}} _ == _ 0

Conversely, if the vector field #~v is solenoidal, then &exist. a vector field ~#u (not unique) such that _ ~#v _ = _ curl ~#u.