       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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