13.1: Vector-Valued Functions

University:
The Pennsylvania State University
Course:
Math 231 | Calculus Severable Variables
Academic year:

2024
Views:

85

Pages:

1
Author:

Alexis H.

13.1 Vector-Valued Functions Consider a particle moving in R^3 whose coordinates at time t are (x (t), y (t), z (t)). It is convenient to represent the particles path by the vector-valued function r (t) = hx (t) ; y (t) ; z (t)i = x (t) i + y (t) j + z (t) k More generally, a vector-valued function is any function r(t) of the form above whose domain is a set of real numbers and whose range is a set of position vectors The variable t is called a parameter, and the functions x (t), y (t), z (t) are called the components or coordinate functions The terminal point of a vector-valued function r(t) traces a path in R^3 as t varies. We refer to r(t) either as a path or as a vector parametrization of a path Such a curve is described equally well by the vector-valued function r(t) = . The difference lies only in whether we visualize the path as traced by a moving point c(t) or the tip of a moving vector r(t) The curve is the set of all points (x (t), y (t), z (t)) as t ranges over the domain of r(t). The path is a particular way of traversing the curve; it may traverse the curve several times, reverse direction, move back and forth, etc. A curve in R^3 is also referred to as a space curve (as opposed to a curve in R^2, which is called a plane curve). Space curves can be quite complicated and difficult to sketch by hand The projections onto the the coordinate planes are another aid in visualizing space curves. The projection of a path r(t) = onto the xy-plane is the path P(t) = . Similarly, the projections onto the yz- and xz-planes are the paths <0, y (t), z (t)> and , respectively Example: parametrizing the intersection of surfaces Parametrize the curve obtained as the part of the intersection of the surfaces x2 + y 2 = 4 First method: x2 + y 2 = 4 y 2 = 4 ¡ x2 ¡ ¢ z = x2 ¡ y 2 + 1 = x2 ¡ 4 ¡ x2 + 1 = 2x2 ¡ 3 r (t) = ht; x2 ¡ y 2 = z ¡ 1 p y=§ p 4 ¡ x2 4 ¡ t2 ; 2t2 ¡ 3i Second method: x = 2 cos t y = 2 sin t z = x2 ¡ y 2 + 1 = 4 cos2 t ¡ 4 sin2 t + 1 = 4 cos 2t + 1 r (t) = h2 cos t; 2 sin t; 4 cos 2t + 1i and