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MathGroup Archive 2004

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multiple derivatives?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg45340] multiple derivatives?
  • From: Michael Alfaro <malfaro at ucsd.edu>
  • Date: Thu, 1 Jan 2004 05:54:32 -0500 (EST)
  • References: <200312271000.FAA02128@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi all,

I have been struggling with the following problem and would be grateful 
for any advice on how to proceed.  The rather ugly function below (uKT) 
describes a four-bar mechanism found in the jaws of some fishes and 
relates lower jaw opening to motion expressed at another jaw bone (uKT 
is a measure of mechanical advantage).

(uDiagonal[input_, fixed_, angleA_] := Sqrt[input^2 + fixed^2 - 
2*input*fixed*Cos[angleA*(Pi/180)]]; )*
    (uAngYZ[fixed_, input_, coupler_, output_, angleA_] :=
      ArcCos[(coupler^2 + output^2 - uDiagonal[input, fixed, 
angleA]^2)/(2*coupler*output)]*(180/Pi); )*
    (uAngW[fixed_, input_, coupler_, output_, angleA_] := 
ArcCos[(uDiagonal[input, fixed, angleA]^2 + fixed^2 - input^2)/
         (2*uDiagonal[input, fixed, angleA]*fixed)]*(180/Pi); )*
    (uAngX[fixed_, input_, coupler_, output_, angleA_] := 
ArcCos[(coupler^2 + uDiagonal[input, fixed, angleA]^2 - output^2)/
         (2*coupler*uDiagonal[input, fixed, angleA])]*(180/Pi); )*
    (uAngY[f_, i_, c_, o_, A_] := 90 - uAngW[f, i, c, o, A] - uAngX[f, 
i, c, o, A]; )*
    (uAngZ[f_, i_, c_, o_, A_] := uAngYZ[f, i, c, o, A] - uAngY[f, i, c, 
o, A]);
uKT[f_, i_, c_, o_, A_] := (uAngZ[f, i, c, o, A + 30] - uAngZ[f, i, c, 
o, A])/30
uKT[1, 0.409923448, 0.420293528, 0.558332627, 30]

I am interested in how uKT changes as a of function to i, c, and o, and 
have been calculating partial derivatives of each of these variables 
for a number of fish species.  Here is an example of the partial 
derivative of i evaluated for one species :

Derivative[0, 1, 0, 0, 0][uKT][1, 0.7`, 0.9`, 0.74`, 30]

However, what I would really like to know is, given a unit small amount 
of change that can be distributed over all three variables in any way,
  how much does uKT change.  I initially thought that I could get at 
that by taking the multiple derivative of all three variables.

Derivative[0, 1, 1, 1, 0][uKT][1, 0.7`, 0.9`, 0.74`, 30]

However, I now have doubts that the multiple partial derivative 
actually tells me this.  Does anyone have any insight into this 
problem?  Thanks!

Michael


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