Calculus of Vector-valued Functions
Differentiation of a vector-valued function
Definition: A vector-valued function for is differentiable if are differentiable on . Then the derivative of is defined as follows:
Equivalently, we can write the derivative in term of limit:
In the applet below, we visualize as a tangent vector in the positive orientation of the curve (if it is not a zero vector).
Exercise: Let for . Then compute the unit tangent vector of the curve parametrized by in terms of .
Properties of differentation
Let and be differentiable vector-valued function, be a differentiable real-valued function, and be a constant vector.
Exercise: Let and . Compute
Exercise: Let be a positive constant and be a vector-valued function such that for any . Prove that is orthogonal to for all . Can you explain this result from a geometry point of view?
Integration of vector-valued functions
An antiderivative of a vector-valued function is if . Explicitly, if and , then , which means that are antiderivative of respectively. Moreover, the indefinite integral of is
where is an arbitrary constant vector.
Example
Find with and
Answer:
Therefore, .
The definite integral of is defined as follows:
By Fundamental Theorem of Calculus, we have
where is an antiderivative of .
Exercise: Suppose the tangent vector of a curve parametrized by for is . Find the vector from the point on the curve when to the point on the curve when .