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
- 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.
- 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.
Ex: and then
- Multiplication by a scalar, :
- Cross Product
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.