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VF.2 Derivatives and Integrals of Vector Functions

The derivative r'(t) of a vector function r(t) is defined in much the same way as for real-valued functions:

 

 

This is an animation that explains the concept:

 

The definition of the derivative is graphically illustrated above  with P and Q being the position vectors of and represents the vector As  h it appears that this vector approaches a vector that lies on the tangent line. For this reason, the vector r'(t) is called the tangent vector to the curve defined by r at the point P. The tangent line to C at P is defined to be the line through P parallel to the tangent vector at point P. Also, to define the unit tangent vector, we use:

The following gives a convenient method for computing the derivative of a vector function r: just differentiate each component of r.

If , where f, g, and h are differentiable function, then :

 

Here is a graph with multiple derivative vectors on it:

Rule for Differentiation

Suppose u and v are differentiable vector functions, c is a scalar, and is a real-valued function. Then:

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Integrals

The definite integral of a continuous vector function r(t) can be defined in much the same way as or real-valued functions except that the integral is a vector. But then we can express the integral of r in terms of the integrals of its component functions f , g, and h as follows:

This means that we can evaluate an integral of a vector function by integrating each component function.

Also, we can extend the Fundamental Theorem of calculus to continuous vector functions as follows:

where R is an antiderivative of r, that is, R'(t)=r (t).     We use the notation for indefinite integrals.

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