Chap13_Sec2

# Chap13_Sec2 - VECTOR FUNCTIONS 13.2 Derivatives and...

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13.2 Derivatives and Integrals f Vector Functions VECTOR FUNCTIONS of Vector Functions In this section, we will learn how to: Develop the calculus of vector functions.

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The derivative r of a vector function is defined in much the same way as for real-valued functions. if this limit exists. DERIVATIVES, TANGENT VECTOR 0 ( ) ( ) '( ) lim h d t h t t dt h + - = = r r r r If the points P and Q have position vectors r ( t ) and r ( t + h ) , then represents the vector r ( t + h ) – r ( t ). This can therefore be regarded as a secant vector. As h 0, 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 , provided: s r ( t ) exists s r ( t ) 0 PQ uuur
If r ( t ) = f ( t ), g ( t ), h ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , where f , g , and h are differentiable functions, then: = UNIT TANGENT VECTOR '( ) ( ) | '( ) | t T t

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Chap13_Sec2 - VECTOR FUNCTIONS 13.2 Derivatives and...

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