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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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