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VECTOR FUNCTIONS OF ONE VARIABLE
VELOCITY AND CURVATURE VECTOR
CHAPTER 11 FIELDS AND STREAMLINES THE
GRADIENT FIELD DIVERGENCE
Vector

Differential

Calculus

11.1 Vector Functions of One Variable

A vector function of one variable is a function of the form F(t) = x(t)i + y(t)j + z(t)k.
This vector function is continuous at t 0 if each component function is continuous at t 0 .

We may think of F(t) as the position vector of a curve in 3-space. For each t for which
the vector is defined, draw F(t) as an arrow from the origin to the point (x(t), y(t), z(t)).This
arrow sweeps out a curve C as t varies. When thought of in this way, the coordinate functions
are parametric equations of this curve.

EXAMPLE 11.1
2
2
H(t) = t i + sin(t)j − t k is the position vector for the curve given parametrically by
2
2
x = t , y = sin(t), z =−t .
Figure 11.1 shows part of a graph of this curve.

F(t)= x(t)i+ y(t)j+ z(t)k is differentiable at t if each component function is differentiable
at t, and in this case

F (t) = x (t)i + y (t)j + z (t)k.
We differentiate a vector function by differentiating each component.
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