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Gradient of a Scalar Field

 
 

Gradient

&phi. _ = _ &phi. ( ~x, ~y, ~z ) _ is a scalar field over &reals.&powthree., then define the #~{gradient} of &phi.:

grad &phi. _ = _ ~#i fract{∂&phi.,∂~x} _ + _ ~#j fract{∂&phi.,∂~y} _ + _ ~#k fract{∂&phi.,∂~z}

Del Operator

we define the operator "del" or "nabla" :

&nabla. _ = _ ~#i fract{∂,∂~x} _ + _ ~#j fract{∂,∂~y} _ + _ ~#k fract{∂,∂~z}

Then we can write _ _ _ grad &phi. _ = _ &nabla. &phi.

Rate of Change in a Direction

If &phi. is a scalar field over &reals.&powthree., ~P is a point with position vector #~r, and ~#s is some other vector. We are interested in the rate of change of &phi. in the direction of #~s, we define this to be

fract{d&phi.,d~#s}( #~r ) _ #:= _ lim{,&epsilon. -> 0} fract{&phi. ( #~r + &epsilon. #~s ) - &phi. ( #~r ), &epsilon. | #~s | } _ _ _ _ _ _ _ _ [ note that this is a vector quantity. ]

Now

&phi. ( #~r + &epsilon. #~s ) - &phi. ( #~r ) _ ~= _ fract{∂&phi.,∂~x} ( #~r ) &epsilon. ~x _ + _ fract{∂&phi.,∂~y} ( #~r ) &epsilon. ~y _ + _ fract{∂&phi.,∂~z} ( #~r ) &epsilon. ~z

where _ ~#s _ = _ ~x #~i + ~y ~#j + ~z ~#k. _ But this is just _ &nabla.&phi. &dot. &epsilon. #~s , _ and is exact in the limit, so

fract{d&phi.,d~#s}( #~r ) _ = _ lim{,&epsilon. -> 0} fract{&nabla.&phi. &dot. &epsilon. #~s, &epsilon. | #~s | } _ = _ &nabla.&phi. &dot. hat{#~s}

where _ hat{#~s} _ is the unit vector in the direction of #~s.