ABSTRACT
Moreover, one can define the derivative rl(t) in the same way as derivatives of scalar functions. Precisely, the derivative rl( t) of the vector-valued function r(t) we call the limit:
If this limit exists, then r(t) is called dzfferentiable. The differentiation of vectorvalued functions has the same properties as the differentiation of scalar functions. The sum of the derivatives of two functions is equal to the derivatives of the sum of these two functions:
i.e. the rule of differentiation in this case is the same as the diffcrentiation rule for products of scalar functions. The differcntiation of inner products, vector products, and mixed products of vector-valued functions is computed by the consecutive differentiation of the cofactors. To be precise, if pl(t) , p2(t), ~ ~ ( 1 ) are vector-valued functions, then
For example, let us prove the differentiation rule for the vector product:
If rl(t) = 0, then r(t) is a constant vector: r(t) = c. By definition, the second derivative rl'(t) of r(t) is the derivative of rl(t). By
This system of three equations can be rewritten as
where o( It - f o l k ) denotes a vector whose length is an infinitesimal with respect to It - tolk as t -, t o We remark that there exists one essential difference between Taylor's expansion of vector-valued functions and Taylor's expansion of scalar functions. If we consider Taylor's expansion for a scalar function , f(t), then we have
The following properties of the defined integral are obvious:
For example, let us prove the second formula. The first component of the vector
is equal to
where a, are the components of the vector a. We see that on the right-hand side of the last equality we have the first component of the vector
Considering the second and third components similarly, we prove the desired formula.
