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Question: Checking a theorem about ultrahyperbolic PDE

Question:Checking a theorem about ultrahyperbolic PDE

Al86 145 Maple
pde
August 03 2015
0

I have the following paper:

http://projecteuclid.org/download/pdf_1/euclid.dmj/1077490637

 

Now I wanted to check Fritz John's claim in the proof of Theorem 1.1, he says that equation (7) can be easily verified for case i=1,k=2.

 

Now at first I tried to calculate by hand, but it's just a lengthy calculation, so now I turned to maple to check its validity, I get that this claim is false, am I wrong in my code? if yes, then how to change it?

 

P.S

I changed between xi and x and eta and y.

In the following is the code:

 

> 

v := (((x_1-y_1)/(x_3-y_3))^2+((x_2-y_2)/(x_3-y_3))^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3)); 1; diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_1), y_2); 1; diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_2), y_1); 1; evalb(diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_1), y_2) = diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_2), y_1))

((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))

  

u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(3/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^4)+(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*(x_1-y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)-u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(-2*x_2+2*y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)+((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(-(D[1, 1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*y_3/(x_3-y_3)-(-(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))/(x_3-y_3)-(-(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*y_3/(x_3-y_3)+(-(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))/(x_3-y_3))/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(-2*x_2+2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_1-2*y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*(2*x_1-2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+2*((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_1-2*y_1)*(-2*x_2+2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^3

  

u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(3/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^4)+((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)-u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_2-y_2)*(-2*x_1+2*y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3))*(x_1-y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)+((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(((D[1, 1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))/(x_3-y_3)-((D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*y_3/(x_3-y_3)-((D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))/(x_3-y_3)-((D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*y_3/(x_3-y_3))/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3))*(-2*x_1+2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_2-2*y_2)*(x_1-y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(2*x_2-2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+2*((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_2-2*y_2)*(-2*x_1+2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^3

  

false

 (1)
> 

``
> 

``

 

Download ultrhyperbolic_pde.mw

> 

v := (((x_1-y_1)/(x_3-y_3))^2+((x_2-y_2)/(x_3-y_3))^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3)); 1; diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_1), y_2); 1; diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_2), y_1); 1; evalb(diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_1), y_2) = diff(diff(v/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2), x_2), y_1))

((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))

  

u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(3/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^4)+(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*(x_1-y_1)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)-u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(-2*x_2+2*y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^2)+((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(-(D[1, 1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*y_3/(x_3-y_3)-(-(D[1, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))/(x_3-y_3)-(-(D[1, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3, 3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*y_3/(x_3-y_3)+(-(D[1, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4, 4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))/(x_3-y_3))/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)-(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)-(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*y_3/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3))*(-2*x_2+2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_1-2*y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2*(x_3-y_3)^2)-((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*(-(D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3))*(2*x_1-2*y_1)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^2+2*((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(1/2)*u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(2*x_1-2*y_1)*(-2*x_2+2*y_2)/((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)^3

  

u((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*(x_1-y_1)*(x_2-y_2)/(((x_1-y_1)^2/(x_3-y_3)^2+(x_2-y_2)^2/(x_3-y_3)^2+1)^(3/2)*((x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2)*(x_3-y_3)^4)+((D[1](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)+(D[2](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))/(x_3-y_3)+(D[3](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2+x_1-y_1)/(x_3-y_3))*x_3/(x_3-y_3)-(D[4](u))((-x_1*y_3+x_3*y_1+x_2-y_2)/(x_3-y_3), (-x_2*y_3+x_3*y_2-x_1+y_1)/(x_3-y_3), (-x_1*y_3+x_3*y_1-x_2+y_2)/(x_3-y_3), 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false

 (1)
> 

``
> 

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