Vector Differentiation

In calculus we compute derivatives of real functions of a real variable. In the case of functions of a single variable

            y = f(x)

we compute the derivative of y with respect to x. In the case of a function of several variables

            y = f(x₁, x₂, ... , x_n)

we can compute the derivative of y with respect to any one of the independent variables, viewing the others as fixed.

In vector analysis we compute derivatives of vector functions of a real variable; that is we compute derivatives of functions of the type

            F(t) = f₁(t) i + f₂(t) j + f₃(t) k

This function can be viewed as describing a space curve. Intuitively it can be regarded as a position vector, expressed as a function of t, that traces out a space curve with increasing values of t. Expressed in different notation it is the function
      R(t) = x(t) i + y(t) j + z(t) k