Bard Math 212 - Vector Valued Functions

557 days ago by ethan

A Vector-Valued Function in \mathbb{R}^3

r(t) = (\cos t, \sin t, t).
t = var('t') parametric_plot3d((cos(t), sin(t), t), (t,0,4*pi), color='purple', aspect_ratio=1) 
       

A Vector-Valued Function in \mathbb{R}^3, with Thicker Plot

r(t) = (t, t^2, t^3).
t = var('t') parametric_plot3d((t, t^2, t^3), (t,-2,2), thickness=2, color='purple', aspect_ratio=1) 
       

Two Surfaces in \mathbb{R}^3 that Intersect in a Curve

The graph of x^2 + y^2 = 1 is in blue.

The graph of x^2 - z = 0 is in yellow.

The intersection, in purple, is the curve given by the vector-valued function r(t) = (\sin t, \cos t, {\sin}^2 t).
x, y, z, t = var('x,y,z,t') P = parametric_plot3d((sin(t), cos(t), (sin(t))^2), (t,-pi,pi), color='purple', thickness=2, aspect_ratio=1) Q = implicit_plot3d(x^2 - z==0, (x, -4, 4), (y, -4, 4), (z, -4, 4), color='yellow', opacity=0.7, aspect_ratio=1) R = implicit_plot3d(x^2 + y^2==1, (x, -4, 4), (y, -4, 4), (z, -4, 4), opacity=0.7, aspect_ratio=1) P + Q + R