8.1. VECTOR FUNCTIONS

If to each value of a scalar variable t, there corresponds a value of a vector , then is called a vector function of the scalar variable t and we write = or = ( t).

For example, the position vector of a particle moving along a curved path is a vector function of time t, a scalar.

Since every vector can be uniquely expressed as a linear combination of three fixed noncoplanar vectors, therefore, we may write where , , denote unit vectors along the axis of x, y, z respectively, f ₁( t), f ₂( t) and f ₃( t) are called the components of the vector ( t) along the coordinate axes.

8.2. DERIVATIVE OF A VECTOR FUNCTION WITH RESPECT TO A SCALAR

Let = ( t) be a vector function of the scalar variable t. Let ? t be a small increment in t and ? , the corresponding increment in .

Then so that

and

If exists, then the value of this limit is denoted by and is called the derivative of with respect to t .

Since is itself a vector function of t, its derivative is denoted by and is called the second derivative of with respect to t . Similarly, we can define higher...