Vector Differentiation

     As with functions of one variable we define the derivative as the limit of a difference quotient.

Definition (Difference Quotient) The difference quotient of a vector function is the vector function:



where is a change of the variable

Definition (Derivative of a Vector Function) The derivative of a vector function is the vector function defined as the limit



provided this limit exists.

Proposition (Derivative of a Vector Function) The vector function



is differentiable whenever the component functions , , and are each differentiable and in this case

    Proof. By the definition of the derivative of a vector function,





Using properties of limits of vector functions and the fact that the component functions , , and are each differentiable we have,  



  

Example (Derivative of a Vector Function) Find the derivative of the vector function



    Solution. The derivative is the vector function  

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Vector Differentiation
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/vector-differentiation.html