Vector Calculus

(1) Example (Find a Vector-Valued Function)  Find a vector-valued function whose graph is the curve of intersection of the hemisphere   and the parabolic cylinder .

    Solution. One way to accomplish the task is by letting Then and   Therefore a value-valued function for this intersection is     which has the following graph.

(2) Example (Find a Vector-Valued Function) Find a vector-valued function whose graph is the curve of intersection of the plane and the plane

    Solution. One way to accomplish the task is by letting Then to find relations for and we will solve the system

.

Eliminating we have, and so   Solving the first for we find



Therefore a vector-valued function for this intersection is   which has the following graph.

(3) 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.

(4) Example (Limits of Vector-Valued Functions) Given and 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 is continuous.

    Solution. The component function is continuous for all real numbers The component function is continuous for all real number, however, is not continuous when and so he function is continuous for all real numbers in its domain which is

(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,

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