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 $\mathcal{R}^n$ for $n=1,2,3$
  - Vector Operations
- Scalar and vector fields
- Gradient $\nabla$ , divergence $\nabla \circ$ , curl $\nabla \times$ and covariant derivatives on fields
- Composite operators such as $\nabla \circ \nabla \bold{v}$

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 $\mathcal{R}$ .

Definition of a scalar:

A scalar is a single real number, $\displaystyle a \in \mathcal{R}$ . For example $3$ is a scalar.

Definition of a real vector:

A real vector, $\displaystyle v$ is an ordered set of two or more real numbers.

For example: $\displaystyle v = (1,2)$ , $\displaystyle w = (5,0,50,-1.25)$ are both vectors. We will use the notation of $\displaystyle v_i$ where the lower index $\displaystyle i = 1..n$ represents the individual elements of a vector in the appropropriate order.

Ex: The vector $\displaystyle v=(3,-7)$ has two elements, the first element is designated $\displaystyle v_1=3$ and the second is $\displaystyle v_2=-7$

Dimension of a vector:

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

Ex: Dimension of $\displaystyle v = (-1.25, 0,-2,-2)$ is $n=4$

Vector Operations:

To refresh your memory, for vectors of the same dimension the following are valid operations:
Let $v = (a,b)$ and $u=(c,d)$ for each of the following statements.

Addition: $\bold{v + w} = (a,b)+(c,d)=(a+c)+(b+d)$

Ex: $v = (2,5)$ and $u=(6,1)$ then $v + w = (8,6)$

Multiplication by a scalar, $\displaystyle k$ : $k ( v ) = k(a,b)=(ka,kb)$

Ex: $k =2$ and $k (-2,4)=2(-2,4)=(-4,8)$

Cross Product $\bold{u} \times \bold{v} = \bold{w}$

Let $\bold{u = (2,3,4) \mbox{ and } v = (-1,4,-3)}$ then
$\bold{u} \times \bold{v} =\left [ \begin{array}{ccc} i&j&k\\ 2&3&4 \\ -1&4&-3 \end{array} \right ] = \bold{i}(3 (-3)- 4^2) - \bold{j} (2 (-3) -4 (-1)) + \bold{k} (2(4)- 3(-1)$
$\bold{u} \times \bold{v} = -25 \bold{i} + \bold{j} + 11 \bold{k}$

Lesson on Scalar and Vector Fields

Lesson on Solving Boundary Value Problems with Nonhomogeneous BCs

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