Vector Calculus Tutorial

(1) Definition (Operations with Vector-Valued Functions) Let and be vector-valued functions of the real variable , and let be a real-valued function. Then and are vector-valued function defined as follows:







These operations are defined on the intersection of the domain of the vector-valued and real-valued functions that occur in the definitions, respectively.

(2) Example (Operations with Vector-Valued Functions)  Given and find the vector function

    Solution. We have
    

    






(3) Example (Operations with Vector-Valued Functions)  Given the vector-valued functions







find


     Solution. We have,









(4) Example (Operations with Vector-Valued Functions)  Given the vector-valued functions







find


     Solution. We have,








  

(5) Definition (Continuity of Vector-Valued Functions) A vector-valued function is continuous at means is in the domain of and . Further, a vector function is continuous on an interval if it is continuous at every point in the interval.

(6) Example (Continuity of Vector-Valued Functions) Determine where the vector-valued function   



where is continuous.  

    Solution. The function is continuous for all real numbers in its domain which is because



and









for all real numbers such that

(7) Proposition (Derivative of a Vector Function) The vector function



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

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

    Solution. The derivative is the vector function  

(9) Example (Tangent Vector) Find a tangent vector at the point where for

    Solution. We have,



and the tangent line to the graph of for is the line that passes through the point and is determined by the parametric equations    and because this line passes through and is parallel to the tangent vector at namely,

(10) Proposition (Derivative Rules for Vector Functions) If the vector functions and the scalar function are differentiable at , and if and are constants, then are differentiable at and,

    (i)  Linearity:  
    
    (ii)  Scalar Multiple:
    
    (iii)  Dot Product:
    
    (iv)  Cross Product:
    
    (v)  Chain Rule:
        

(11) Example (Derivative Rules for Vector Functions) Compute the derivative of the vector function given by where

    and     

    Solution. We have,

(12) Definition (Indefinite Integral of a Vector Function) Let  

,

where , , and are continuous on the closed interval Then the indefinite integral of is

(13) Example (Indefinite Integral of a Vector Function) Compute

    Solution. We have,





where is a constant vector. Not that we used using integration by parts where is a constant.

(14) Definition (Definite Integral of a Vector Function) Let

,

where , , and are continuous on the closed interval Then the definite integral of is the vector  

(15) Example (Definite Integral of a Vector Function) Given the vector function



Find the value of for which

    Solution. We have,
    
  


  
Thus, and So,

(16) Definition (Smooth Curve) The graph of the vector function defined by is smooth on any interval of where is continuous and .

(17) Example (Smooth Curve) Determine where the graph of the vector function



is smooth.

    Solution. The graph of the vector function is smooth over any interval not containing because



and for any except

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Cite this as:
Vector Calculus Tutorial
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/vector-calculus-tutorial.html