# Boundary Value Problems/Lesson 5.1

## 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.

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

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 $\mathbf {u} \times \mathbf {v} =\mathbf {w}$

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