Computational Geometry Seminar

Wichita State University, Dept. of Mathematics

Vector fields along space curves

Last changed: 3 Oct 2017

Authors: Jen Pinkston, Justin Ryan, Arelis Silva

%display latex

# define the curve
var('t x y z');
a(t) = [cos(t),sin(t),2*t];
V(t) = [t,1-t^2,1+t^2];
C = matrix([[-1,0,0],
            [0,1/sqrt(2),-1/sqrt(2)],
           [0,1/sqrt(2),1/sqrt(2)]]);

A = -pi;
n = 10;
tp = 2*A/(n-1);

p = {}
for i in range(n):
    p[i] = [a[0].substitute(t = A+tp*i),a[1].substitute(t = A+tp*i),a[2].substitute(t = A+tp*i)];

VV = {}
for i in range(n):
    VV[i] = [V[0].substitute(t=A+tp*i),V[1].substitute(t=A+tp*i),V[2].substitute(t=A+tp*i)];
    
var = {}
for i in range(n):
    var[i] = plot((1/3)*vector(VV[i]), start=vector(p[i]), color='blue', width='1.5');

vector_plot = sum(var[i] for i in range(n));

graph = parametric_plot3d(a,(t,-4*pi,4*pi),color='red');
show(graph + vector_plot)

Cp = {}
for i in range(n):
    Cp[i] = C*vector([a[0].substitute(t = A+tp*i),a[1].substitute(t = A+tp*i),a[2].substitute(t = A+tp*i)]);
                        
CVV = {}
for i in range(n):
    CVV[i] = C*vector([V[0].substitute(t=A+tp*i),V[1].substitute(t=A+tp*i),V[2].substitute(t=A+tp*i)]);
    
Cvar = {}
for i in range(n):
    Cvar[i] = plot((1/3)*vector(CVV[i]), start=vector(Cp[i]), color='purple', width='1.5');

C_vector_plot = sum(Cvar[i] for i in range(n));

Ca = C*vector(a(t));
C_graph = parametric_plot3d(Ca,(t,-4*pi,4*pi),color='green');

show(graph + C_graph,aspect_ratio=1)

show(graph + C_graph + vector_plot + C_vector_plot, aspect_ratio=1)

%%octave

B = randi(10,3,3);
B

B =

 5 7 3
 9 4 2
 10 3 9

B = matrix([[5,1,2],
            [3,4,3],
            [7,3,10]]);

B.det()

$$\newcommand{\Bold}[1]{\mathbf{#1}}108$$

B = B/(det(B))^(1/3);

B

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} \frac{5}{12} \cdot 4^{\frac{2}{3}} & \frac{1}{12} \cdot 4^{\frac{2}{3}} & \frac{1}{6} \cdot 4^{\frac{2}{3}} \\ \frac{1}{4} \cdot 4^{\frac{2}{3}} & \frac{1}{3} \cdot 4^{\frac{2}{3}} & \frac{1}{4} \cdot 4^{\frac{2}{3}} \\ \frac{7}{12} \cdot 4^{\frac{2}{3}} & \frac{1}{4} \cdot 4^{\frac{2}{3}} & \frac{5}{6} \cdot 4^{\frac{2}{3}} \end{array}\right)$$

det(B)

$$\newcommand{\Bold}[1]{\mathbf{#1}}1$$

B = matrix([[1,2,3],
           [3,2,1],
           [-1,1,-1]]);
det(B)

$$\newcommand{\Bold}[1]{\mathbf{#1}}16$$

Bp = {}
for i in range(n):
    Bp[i] = B*vector([a[0].substitute(t = A+tp*i),a[1].substitute(t = A+tp*i),a[2].substitute(t = A+tp*i)]);
                        
BVV = {}
for i in range(n):
    BVV[i] = B*vector([V[0].substitute(t=A+tp*i),V[1].substitute(t=A+tp*i),V[2].substitute(t=A+tp*i)]);
    
Bvar = {}
for i in range(n):
    Bvar[i] = plot((1/3)*vector(BVV[i]), start=vector(Bp[i]), color='cyan', width='1.5');

B_vector_plot = sum(Bvar[i] for i in range(n));

Ba = B*vector(a(t));
B_graph = parametric_plot3d(Ba,(t,-4*pi,4*pi),color='brown');

show(graph + C_graph + B_graph,aspect_ratio=1)

show(graph + B_graph + vector_plot + B_vector_plot,aspect_ratio = 1)

B = matrix([[t,0,0],
       [0,t^2,0],
       [0,0,1-t]]);

B

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} t & 0 & 0 \\ 0 & t^{2} & 0 \\ 0 & 0 & -t + 1 \end{array}\right)$$

B_star = matrix([[1,0,0],
                [0,2*t,0],
                [0,0,-1]]);
B_star

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 2 \, t & 0 \\ 0 & 0 & -1 \end{array}\right)$$

Bp = {}
for i in range(n):
    Bp[i] = B(t = A+tp*i)*vector([a[0].substitute(t = A+tp*i),a[1].substitute(t = A+tp*i),a[2].substitute(t = A+tp*i)]);
                        
BVV = {}
for i in range(n):
    BVV[i] = B_star(t = A+tp*i)*vector([V[0].substitute(t=A+tp*i),V[1].substitute(t=A+tp*i),V[2].substitute(t=A+tp*i)]);
    
Bvar = {}
for i in range(n):
    Bvar[i] = plot((1/3)*vector(BVV[i]), start=vector(Bp[i]), color='cyan', width='1.5');

B_vector_plot = sum(Bvar[i] for i in range(n));

Ba = B*vector(a(t));
B_graph = parametric_plot3d(Ba,(t,-4*pi,4*pi),color='brown');

show(graph + B_graph + vector_plot + B_vector_plot)

f1 = x^2;
f2 = y^3;
f3 = sqrt(z^2 + 1);

F = vector((f1,f2,f3))

F

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(x^{2},\,y^{3},\,\sqrt{z^{2} + 1}\right)$$

Fx = F.diff(x)

Fy = F.diff(y)

Fz = F.diff(z)

JF = matrix([[Fx[0],Fy[0],Fz[0]],
             [Fx[1],Fy[1],Fz[1]],
             [Fx[2],Fy[2],Fz[2]]]);

JF

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 2 \, x & 0 & 0 \\ 0 & 3 \, y^{2} & 0 \\ 0 & 0 & \frac{z}{\sqrt{z^{2} + 1}} \end{array}\right)$$

p = (1,2,3);
V = vector([2,-3,1]);

Fp = F(x=1,y=2,z=3)

JF_p = JF(x=1,y=2,z=3);
JF_p

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 12 & 0 \\ 0 & 0 & \frac{3}{10} \, \sqrt{10} \end{array}\right)$$

VFp = JF_p*V

show(plot(point((1,2,3),size=10,color='red'))+ plot(V,start=vector(p),color='blue') +
     plot(point(Fp,size=10,color='purple')) +plot(VFp,start=vector(Fp),color='green') )

jacobian(F,(x,y,z))

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 2 \, x & 0 & 0 \\ 0 & 3 \, y^{2} & 0 \\ 0 & 0 & \frac{z}{\sqrt{z^{2} + 1}} \end{array}\right)$$