properties of vector-valued functions
If and are vector-valued and a real-valued function of the real variable , one defines the vector-valued functions and componentwise as
and the real valued dot product as
If , one my define also the vector-valued cross product function as
are valid, in additionally
Likewise one can verify the following theorems.
Theorem 1. If is continuous in the point and in the point , then
is continuous in the point . If is differentiable in the point and in the point , then the composite function is differentiable in and the chain rule
is in .
Theorem 2. If and are integrable on , so is also , where are real constants, and
Theorem 3. Suppose that is continuous on the interval and . Then the vector-valued function
is differentiable on and satisfies .
Theorem 4. Suppose that is continuous on the interval and is an arbitrary function such that on this interval. Then
Theorem 2 may be generalised to
Theorem 5. If is integrable on and is an arbitrary vector of , then dot product is integrable on this interval and
|Title||properties of vector-valued functions|
|Date of creation||2013-03-22 19:02:42|
|Last modified on||2013-03-22 19:02:42|
|Last modified by||pahio (2872)|