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Unformatted text preview: Section 14.2 Derivatives and Integrals of Vector Functions “Calculus for vectors” In this section, we generalize the ideas from Calc 1 and 2 to vector valued functions. 1. Derivatives We define the derivative of a vector function in exactly the same way as a scalar valued function - using a difference quotient. Definition 1.1. We define the derivative of a vector function vector r ( t ) and denote it as follows: dvector r dt = vector r ′ ( t ) = lim h → vector r ( t + h ) − vector r ( t ) h provided the limit exists. If the limit exists everywhere, the curve is said to be smooth. The value of the derivative for scalar valued functions coincides with the slope of a graph at a point P . There is a similar interpretation for vector valued functions. Recall that a vector valued function defines a curve C in 3-space (by placing the tail of each vector at the origin- position vectors). Under this realization, for any value of t at the point P = vector r ( t ), the derivative vector vector r ′ ( t ) points in the direction of the tangent line of the curve C at P (see illustration below). (r(t+h)-r(t))/h (t,r(t)) r(t) r(t+h) For this reason, we call the derivative vector r ′ ( t ) the tangent vector of C provided it exists and vector r ′ ( t ) negationslash = vector 0. We also define the tangent line to be the line which passes through P and points in the direction of vector r ′ ( t ). Since vectors can have varying length, there are in fact infinitely many different tangent vectors to a curve ar a given point. Therefore, to avoid ambiguity, we often consider tangent vectors with length 1. Specifically, we define the following: Definition 1.2. We define the unit tangent vector to a vector function vector r ( t ) to be the vector vector T ( t ) defined by vector T ( t ) = vector r ′ ( t ) | vector r ′ ( t ) | ....
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