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# nabla?

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i am not familiar with advanced vector algebra stuff because i am not studying yet but i want to understand it right now because i need it and i think it's quite interesting the nabla-operator (i'll write it as 'V') (i mean that vertically flipped delta aka anti-delta) is afaik a partial derivation in each direction ie: V a = ( da/dx; da/dy; da/dz ) where a is a vector (R^3) but i frequently encounter sth like this V * u = 0 where u is the speed of a particle (in smoke or fluids) explanation: "this equation states that the velocity should conserve mass" you guessed already what my first question is ;-) WHY? 2. what's the product of the nabla-operator and a vector ? 3. whats the cross-product of the nabla-operator and another nabla-operator ? 4. does V^n (nabla_op^n) mean the nth derivation? e.g V^3 a = ( d^3a/dx^3; d^3a/dy^3; d^3a/dz^3 ) ? thanks in advance! [edited by - diego_rodriguez on April 20, 2002 10:55:29 AM]

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I can''t answer all of these questions, but I''m pretty sure that V * u = 0 means that the dot product of the gradient vector (that''s how we call it in french, anyway) with your particle''s speed is 0, so they are both perpendicular. If I am not mistaken, moving perpendicularly to the gradient vector is moving along the level surfaces (again, direct translation from french). Level surfaces are what you see on topographical maps, except in 3D. So if your function represents energy as a function of position, moving perpendicularly to V implies that the derivative of the function in that direction is null, so there is no energy gain. Still, you didn''t mention what your functions represent, so we can''t tell what the cross product of two V will be. A V is simply a vector in the direction where the derivative is maximized.

Could anyone write the proper translation for gradient vector (nabla operator? it sounds weird) and level surfaces?

Cédric

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You should check documents related to electromagnetism : this domain is the root of many domains such as fluids. As I'm french, I'm not sure I should translate my answer because it won't be 'international'. AFAIK, nabla is an english operator (written as a reversed capital delta) available only for cartesian coordinates :
_   d  _    d  _    d  _                         _V = -- ux + -- uy + -- uz (d:partial derivative, X:vectors)    dx      dy      dz
It make no sense if not used with something.

The "gradient" of a vectorial function f is (in cartesian):
_    df _    df _    df _ Vf = -- ux + -- uy + -- uz     dx      dy      dz
The "divergence" of a vectorial quantity A is (in cartesian):
_ _   dAx   dAy   dAzV.A = --- + --- + ---      dx    dy    dz
The "rotationnel" of a vectorial quantity A is :
_ _    dAz   dAy  _      dAx   dAz  _      dAy   dAx  _V^A =( --- - --- )ux + ( --- - --- )uy + ( --- - --- )uz       dy    dz          dz    dx          dx    dy
There is an other operator written as capital delta aka 'Laplacien'...

...but I'll try to find a good ressource in english!

[edited by - Bloodscourge on April 20, 2002 11:32:25 AM]

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it seems that :

french -> english

divergence -> divergence
rotationnel -> curl

more here and there!!!

[edited by - Bloodscourge on April 20, 2002 12:05:17 PM]

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As in, Vf is "del f".
Its a synonym for "grad f"

Its not a stand alone symbol-
Its a operator like d/dx

Taken from my calc book(american)

Bugle4d

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I''m closing this thread because it hasn''t really focused on game development.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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