Chapter 12

Vector Valued Functions

12.1 Vector Valued FunctionsVector valued functions have values in Rp where p is an integer at least as large as 1. Hereare some examples.

Example 12.1.1 A rocket is launched from the rotating earth. You could define a functionhaving values in R3 as (r (t) ,θ (t) ,φ (t)) where r (t) is the distance of the center of massof the rocket from the center of the earth, θ (t) is the longitude, and φ (t) is the latitude ofthe rocket.

Example 12.1.2 Let f (x,y)=(sinxy,y3 + x,x4

). Then f is a function defined on R2 which

has values in R3. For example, f (1,2) = (sin2,9,16).

As usual, D(f) denotes the domain of the function f which is written in bold face be-cause it will possibly have values in Rp. When D(f) is not specified, it will be understoodthat the domain of f consists of those things for which f makes sense.

Example 12.1.3 Let f (x,y,z) =(

x+yz ,√

1− x2,y)

. Then D(f) would consist of the set of

all (x,y,z) such that |x| ≤ 1 and z ̸= 0.

There are many ways to make new functions from old ones.

Definition 12.1.4 Let f ,g be functions with values in Rp. Let a,b be points of R (scalars).Then af +bg is the name of a function whose domain is D(f)∩D(g) which is defined as

(af +bg)(x) = af (x)+bg (x) .

f ·g or (f ,g) is the name of a function whose domain is D(f)∩D(g) which is defined as

(f ,g)(x)≡ f ·g (x)≡ f (x) ·g (x) .

If f and g have values in R3, define a new function f ×g by

f ×g (t)≡ f (t)×g (t) .

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