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 $$ \nabla \circ \nabla \mathbf{v} $$


 *  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. Ex: $$ v = (2,5) $$ and $$ u=(6,1) $$ then $$ v + w = (8,6) $$ Ex: $$ k =2 $$ and $$ k (-2,4)=2(-2,4)=(-4,8) $$ Let $$ \mathbf{u = (2,3,4) \mbox{ and } v = (-1,4,-3)} $$ then $$ \mathbf{u} \times \mathbf{v} =\left [ \begin{array}{ccc} i&j&k\\ 2&3&4 \\ -1&4&-3 \end{array} \right ] = \mathbf{i}(3 (-3)- 4^2) - \mathbf{j} (2 (-3) -4 (-1)) + \mathbf{k} (2(4)- 3(-1)$$ $$ \mathbf{u} \times \mathbf{v} = -25 \mathbf{i} + \mathbf{j} + 11 \mathbf{k} $$
 * Addition: $$ \mathbf{v + w} = (a,b)+(c,d)=(a+c)+(b+d) $$
 * Multiplication by a scalar, $$ \displaystyle k $$: $$ k ( v ) = k(a,b)=(ka,kb) $$
 * Cross Product $$ \mathbf{u} \times \mathbf{v} = \mathbf{w} $$

Lesson on Solving Boundary Value Problems with Nonhomogeneous BCs

 * Watch [[File:Temp.ogg]].