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third

Run Example
(%i1)Q(t):=a+b*t+c*t**2;
                                                 2
(%o1)                       Q(t) := a + b t + c t
(%i2) I(t):=''(diff(Q(t), t));
(%o2)                          I(t) := 2 c t + b
(%i3) s1:first(solve([Q(0)=xm1, Q(h)=x0, I(2*h)=xp1], [a, b, c]));
                          h xp1 + 4 xm1 - 4 x0      h xp1 + xm1 - x0
(%o3)     [a = xm1, b = - --------------------, c = ----------------]
                                  3 h                        2
                                                          3 h
(%i4)  a:second(first(s1));
(%o4)                                 xm1
(%i5) b:second(second(s1));
                              h xp1 + 4 xm1 - 4 x0
(%o5)                       - --------------------
                                      3 h
(%i6) c:second(third(s1));
                               h xp1 + xm1 - x0
(%o6)                          ----------------
                                        2
                                     3 h
(%i7) Q(2*h);
              2 (h xp1 + 4 xm1 - 4 x0)   4 (h xp1 + xm1 - x0)
(%o7)       - ------------------------ + -------------------- + xm1
                         3                        3
(%i8) 
Run Example
display2d:false;

(%o1) false
(%i2) 
Q(t):=a + b*t + c*t*t;

(%o2) Q(t):=a+b*t+c*t*t
(%i3) 
I(t):=''(diff(Q(t), t));

(%o3) I(t):=2*c*t+b
(%i4) 
eq:first(solve([Q(0)=Q2, Q(h2)=Q1, Q(h1+h2)=Q0], [a, b, c]));

(%o4) [a = Q2,
       b = -((2*h1*h2+h1^2)*Q2+(-h2^2-2*h1*h2-h1^2)*Q1+h2^2*Q0)
         /(h1*h2^2+h1^2*h2),c = (h1*Q2+(-h2-h1)*Q1+h2*Q0)/(h1*h2^2+h1^2*h2)]
(%i5) 
a:second(first(eq));

(%o5) Q2
(%i6) 
b:second(second(eq));

(%o6) -((2*h1*h2+h1^2)*Q2+(-h2^2-2*h1*h2-h1^2)*Q1+h2^2*Q0)/(h1*h2^2+h1^2*h2)
(%i7) 
c:second(third(eq));

(%o7) (h1*Q2+(-h2-h1)*Q1+h2*Q0)/(h1*h2^2+h1^2*h2)
(%i8) 
I(h1+h2);

(%o8) 2*(h2+h1)*(h1*Q2+(-h2-h1)*Q1+h2*Q0)/(h1*h2^2+h1^2*h2)
       -((2*h1*h2+h1^2)*Q2+(-h2^2-2*h1*h2-h1^2)*Q1+h2^2*Q0)/(h1*h2^2+h1^2*h2)
(%i9) 
diff(I(h1+h2), Q0);

(%o9) 2*h2*(h2+h1)/(h1*h2^2+h1^2*h2)-h2^2/(h1*h2^2+h1^2*h2)
(%i10) 
Run Example
Vphi = 0;
(%o1)                              Vphi = 0
(%i2) Vt: Vinf(-1+((3/4)*(R/r))+((1/4)*(R/r)^3))*sin(theta);
                                          3
                                         R     3 R
(%o2)                   sin(theta) Vinf(---- + --- - 1)
                                           3   4 r
                                        4 r
(%i3) Vr: Vinf(1-((3/2)*(R/r))+((1/4)*(R/r)^3))*cos(theta);
                                          3
                                         R     3 R
(%o3)                   cos(theta) Vinf(---- - --- + 1)
                                           3   2 r
                                        4 r
(%i4) first: (1/r^2)*diff((r^2)*Vr,r);
                                 3
        2             d         R     3 R
(%o4) (r  cos(theta) (-- (Vinf(---- - --- + 1)))
                      dr          3   2 r
                               4 r
                                                              3
                                                             R     3 R        2
                                      + 2 r cos(theta) Vinf(---- - --- + 1))/r
                                                               3   2 r
                                                            4 r
(%i5) second: (1/(r*sin(theta)))*diff(sin(theta)*Vt,theta);
                                           3
                                          R     3 R
                       2 cos(theta) Vinf(---- + --- - 1)
                                            3   4 r
                                         4 r
(%o5)                  ---------------------------------
                                       r
(%i6) third: 0;
(%o6)                                  0
(%i7) gradv: first+second+third;
                                 3
        2             d         R     3 R
(%o7) (r  cos(theta) (-- (Vinf(---- - --- + 1)))
                      dr          3   2 r
                               4 r
                                                                  3
                                                                 R     3 R
                                              2 cos(theta) Vinf(---- + --- - 1)
                          3                                        3   4 r
                         R     3 R        2                     4 r
  + 2 r cos(theta) Vinf(---- - --- + 1))/r  + ---------------------------------
                           3   2 r                            r
                        4 r
(%i8) 

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