MATHEMATICS 2Q04: Pictures and animations
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ANIMATIONS
- Click here
to see an animation of the helix parametrized by r(t)=(cos t, sin t, t) lying
inside the cylinder x^2+y^2=1.
- Click here
to see an animation of curve r(t)= ((1+cos(2t))/2, sin t) with the derivative
r'(t) shown as a red arrow.
- Click here
to see an animation of the curve r(t)=(t, t^2/2, t^3/3).
- Click here
to see an animation of the helix r(t)=(t, 3 sin(t), 3 cos(t)).
- Click here
to see an animation of a particle moving on the sphere x^2+y^2+z^2=1. Note how the velocity
vector (in red) is always tangent to the sphere as the particle is moving and is thus
always orthogonal to the position vector of the particle.
- Click here
to see an animation of a particle moving under the influence of a force on the helix
r(t)=(sin t, cos t,t) for 0 < t< pi. At time t=pi, the force is switched off and the
particle start moving on a straight line with constant velocity. Its position fot t> pi
is given by r(t)=(pi-t,-1,t).
- Click here
to see an animation of the helix r(t)=(t, 3 sin(t), 3 cos(t)) where the unit tangent
vector (in green), the principal normal vector ( in red ) and the binormal vector (in black)
are shown.(The parameter t is "time").
- Click here
to see an animation of the point describing a circle with radius one with constant angular
speed: r(t)=(cos(t), sin(t). The velocity vector (in red) and acceleration vector (in green) are
shown.
- Click here
to see an animation of the point describing a circle with radius one with a varying angular
speed: r(t)=(cos(t+0.2 sin(3 t)), sin(t+0.2 sin(3 t)). The velocity vector (in red) and acceleration vector ( in green) are
shown.
- Click here
to see an animation of the point describing the curve: r(t)=(arctan(t), log(1+t^2)/2). The velocity vector (in red) and acceleration vector (in green) are
shown.
- Click here to see an animation of the level curves f(x,y)=x^2/4+y^2=A,
where A varies from 0 to 10. Some gradient vectors are plotted on the curve. Note how each gradient vector is orthogonal
to the level curve at the point considered.
- Click here to see an animation
of the first standing wave (for the wave equation).
- Click here to see an animation
of the second standing wave (for the wave equation).
- Click here to see an animation
of the solution of the wave equation u_{tt}=u_{xx} satisfying the boundary conditions u(0,t)=u(pi,t)=0
and the initial conditions u(x,0)= sin(x) and u_t(x,0)=sin(2x).
- Click here to see an animation
of the solution of the heat equation u_t=u_xx satisfying the non-homogeneous booundary
conditions u(0,t)=u_0 and u(L,t)=u_1. Notice how the solution approaches the steady
state solution as t goes to infinity.
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PICTURES
- Click here to see the curve
of intersection of the paraboloid z=9-x^2-y^2 and the plane y=2x.
- Click here to see the curve
r(t)=(t^2,t^2-t,-7t) and its tangent line (in red) at the point (9,6,-21).
- Click here to see the graph of z=sqrt(x^2+y^2-1)
- Click here to see the graph of z=3-2 x-y
- Click here to see the graph of z=sqrt(4-x^2-y^2)
- Click here to see some level curves of f(x,y)=xy.
- Click here to see some level curves of f(x,y)=x^2/4 +y^2.
- Click here the sphere
F(x,y,z)=x^2+y^2+z^2=4, the gradient of F at the point (-1,1,sqrt(2)) on the sphere (in red)
and the plane tangent to the sphere at (-1,1,sqrt(2)).
- Click here to see the vector
field F(x,y,z)=y i.
- Click here to see the vector
field F(x,y,z)=y j.
- Click here to see
a picture of the parametric surface S parametrize by by x=u^-v^2, y=u^2+v, z=uv. Note that
the parametrization is not smooth when (u,v)=(0,0) which corresponds to the point
(0,0,0) on th surface.
- Click here to see the region between the
graphs of z=-y^2 and z=x^2 defined for (x,y) inside the triangle with vertices at (0,0), (1,0) and
(1,1) in the x,y plane.
- Click here to see the region bounded
by the circular paraboloids z=3-x^2-y^2 and z=-5+x^2+y^2 for x,y>0.
- Click here to see the solid region bounded
by the graphs of z=-sqrt(1-x^2-y^2) and z=1 for x^2+y^2<1.
