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

• 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 $ℛ$.

Definition of a scalar:

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

Definition of a real vector:

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

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

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

Dimension of a vector:

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

Ex: Dimension of $v=(-1.25,0,-2,-2)$ 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: $𝐯\mathbf{+}𝐰=(a,b)+(c,d)=(a+c)+(b+d)$

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

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

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

• Cross Product $𝐮\times 𝐯=𝐰$
  

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

Lesson on Scalar and Vector Fields

Lesson on Solving Boundary Value Problems with Nonhomogeneous BCs

• Watch File:Temp.ogg.