# Boundary Value Problems/Lesson 5.1

Click here to return to BVPs Boundary Value Problems

## Lesson Plan[edit | edit source]

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

- Vectors,

**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[edit | edit source]

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

###### Definition of a scalar:[edit | edit source]

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

###### Definition of a real vector:[edit | edit source]

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:[edit | edit source]

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

Ex: Dimension of is

###### Vector Operations:[edit | edit source]

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[edit | edit source]

### Lesson on Operations on Scalar and Vector Fields[edit | edit source]

### Lesson on Solving Boundary Value Problems with Nonhomogeneous BCs[edit | edit source]

- Watch File:Temp.ogg.