Boundary Value Problems/Lesson 5.1

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Lesson Plan


Requirements of student preparation: The student needs to have worked with vectors. If not the student should obtain suitable instruction in vector calculus.

  • Subject Area: A review of vectors, vector operations, the gradient, scalar fields ,vector fields, curl, and divergence.
  • Objectives: The learner needs to understand the conceptual and procedural knowledge associated with each of the following
    • Vectors,
      • Definition of vectors in   for  
      • Vector Operations
    • Scalar and vector fields
    • Gradient  , divergence  , curl   and covariant derivatives on fields
    • Composite operators such as  
  • Activities: These structures are to help you understand and aid long-term retention of the material.
    • Lesson on Vectors, their associated properties and operations that use vectors.
    • Lesson on Scalar and Vector fields
    • Lesson on Operations on scalar and vector fields
  • Assessment: These items are to determine the effectiveness of the learning activities in achieving the lesson objectives.
    • Worksheets
    • Quizzes
    • Challenging extended problems.
    • Student survey/feedback
    • Web analytics

Lesson on Vectors


We will be using only real numbers in this course. The set of all real numbers will be represented by  .

Definition of a scalar:

A scalar is a single real number,  . For example   is a scalar.

Definition of a real vector:

A real vector,   is an ordered set of two or more real numbers.

For example:   ,   are both vectors. We will use the notation of   where the lower index   represents the individual elements of a vector in the appropropriate order.

Ex: The vector   has two elements, the first element is designated   and the second is  

Dimension of a vector:

The dimension of a vector is the number of elements in the vector.

Ex: Dimension of   is  

Vector Operations:

To refresh your memory, for vectors of the same dimension the following are valid operations:
Let   and   for each of the following statements.

  • Addition:  

Ex:   and   then  

  • Multiplication by a scalar,  :  

Ex:   and  

  • Cross Product  

Let   then

Lesson on Scalar and Vector Fields


Lesson on Operations on Scalar and Vector Fields


Lesson on Solving Boundary Value Problems with Nonhomogeneous BCs