The Derivative in Graphing and Applications
Three-Dimensional Space; Vectors
and has sides
of length 1 that are parallel to the coordinate planes.
(b) The second sphere is circumscribed about the cube in part (a).
Vector-Valued Functions
- circular
helix
-
tricuspoint
-
twisted cubic
-
the graph of the vector-valued function
r(t)=(1-t, 3t, 2t) is a line
-
two hyperbolas
in parallel displacement
- The degree measure of the angle between the
tangent lines to the graphs of p(t)=(atan(t), sin(t), t²) and
q(t)=(t² - t, 2t - 2, ln(t))
at the origin
-
The coordinate system determined by T(t), N(t
), and B(t) is called the TNB-frame or sometimes the Frenet
frame in honor of the French mathematician Jean Frédéric Frenet
(1816–1900) who pioneered its application to the study of space curves.
- The curvature
of the ellipse r = 2 cos t i + 3 sin t j (0 ≤ t ≤ 2π)
-
An example of finding arc length
parametrizations
-
Curve((2cos(t), 4sin(t), t), t, 0, 2π)
- Another helix: Curve((3cos(t), sin(t),
0.01.t²), t, 0, 6π)
fig1,
fig2,
fig3
- A
Surface(t, s, sin(t), t, -π, π, s, -π, π)
and a
Curve((t, t, sin(t)), t, -π, π)
on it
fig1,
fig2
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