Increment and Differential

We shall now discuss the increment and differential of a vector function. Given a curve

X = F(t),

f is a scalar independent variable and X a vector dependent variable. We introduce a new scalar independent variable Δt and a new vector dependent variable ΔX with the equation

ΔX = F(t + Δt) - F(t).

ΔX is called the increment of X. ΔX depends on both t and Δt, and is the vector from the point on the curve at t to the point on the curve at t + Δt (Figure 10.8.4).

Figure 10.8.4 The Increment of X

Now suppose the vector derivative F'(t) exists. We introduce another vector dependent variable dX with the equation

dX = F'(t) Δt.

dX is called the differential of X. It is customary to write dt for Δt, so we get the familiar quotient formulas

The relationship between the vector increment and differential may be summarized as follows. At each value t where F'(f) exists and is not zero, and for each nonzero infinitesimal Δt, we have:

dX is an infinitesimal vector tangent to the curve X = F(f). ΔX is an infinitesimal vector which is almost parallel to dX.

dX and ΔX are infinitely close compared to Δt, i.e,

as shown in Figure 10.8.5.

Figure 10.8.5