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

###### Definition of a scalar:

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

###### Definition of a real vector:

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

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

Ex: The vector ${\displaystyle \displaystyle v=(3,-7)}$ has two elements, the first element is designated ${\displaystyle \displaystyle v_{1}=3}$ and the second is ${\displaystyle \displaystyle v_{2}=-7}$

###### Dimension of a vector:

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

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

###### Vector Operations:

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

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

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

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

Ex: ${\displaystyle k=2}$ and ${\displaystyle k(-2,4)=2(-2,4)=(-4,8)}$

• Cross Product ${\displaystyle \mathbf {u} \times \mathbf {v} =\mathbf {w} }$

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