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# Vector-valued functions.

When a particle moves in the space, during a time-interval I, we think of its coordinates as functions of the variable t (= time), namely (x=x(t),y=y(t),z=z(t)). The set of all the locations of the particle is a curve in the space, named the trajectory of the particule, and the functions x,y,z define a parametrization of .

The vector is the position vector of the particle at time t, and the point M such that is the position of the particle at time t.

Without the z-component, we have the corresponding definitions for a curve/ the motion of a particule in the plane.

Example 2.1   Take

We have: , i.e. the trajectory of the particle is on the parabola whose equation is y=1-2x2. Is the trajectory the whole parabola? No! because . With the notations of Figure  2 the trajectory is the arc AC, on which the particule travels alternatively in both directions, with a period of .

Example 2.2   The curve defined by is called a Lissajoux curve'' (it is displayed in Figure  3).

Example 2.3   Take now

The trajectory is the unit circle in the plane whose equation is z=1.

Example 2.4   Take now

The trajectory is a circular helix, whose projection onto the xy-plane is the unit circle and the axis is the z-axis, as in Figure  5.

Next: Limits and Continuity. Up: Curves in the plane Previous: Curves in the plane
Noah Dana-Picard
2001-05-30