Boundary Value Problems/Lesson 5.1
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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.