Calculus III

 

  1. 1. Course Information:

a) Course Description

In this free course, Vectors, we look at some of the equations that represent points, lines and planes in mathematics. We explore concepts such as Euclidean space, vectors, dot and cross products.

b) Course Objectives

The course aims for students is to developing the notion of vectors and their properties in the three-dimensional space and the plane. Presenting the calculus of vector functions and curves.

Providing students with the skills of multiple integration for function of several variables and its application to practical problems.

C) Course Learning Outcomes

By the end of the course, the students should be able to Recognize the 3-space in different types of coordinates systems. Do operations on vectors. Identify different types of equations of lines, planes and surfaces. Recognize different types of calculus operations of vector-valued functions. Find arc length, unit tangent and normal vectors.

           2. Course content Vectors Definition:

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

Two examples of vectors are those that represent force and velocity. Both force and velocity are in a particular direction. The magnitude of the vector would indicate the strength of the force or the speed associated with the velocity.

Vector symbol is or ,We denote the magnitude of the vector .

Types of Vectors

  • Zero Vector: A vector whose initial and terminal points coincide, is called a zero vector (or null vector), and denoted as Zero vector cannot be assigned a definite direction as it has zero magnitude. Or, alternatively otherwise, it may be regarded as having any direction. The vectors represent the zero vector.
  • Unit Vector: A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector in the direction of a given vector is denoted by .
  • Coinitial Vectors: Two or more vectors having the same initial point are called coinitial vectors.
  • Collinear Vectors: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and direction
  • Equal Vectors: Two vectors are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written as .
  • Negative of a Vector: A vector whose magnitude is the same as that of a given vector but direction is opposite to that of it, is called negative of the given vector.

For example, vector is negative of the vector ,and written as = –

Vectors in the plane

We assume that you are familiar with the standard (x,y) Cartesian coordinate system in the plane. Each point p in the plane is identified with its x and y components: p=(p1,p2).

To determine the coordinates of a vector a in the plane, the first step is to translate the vector so that its tail is at the origin of the coordinate system. Then, the head of the vector will be at some point (a1,a2) in the plane. We call (a1,a2) the coordinates or the components of the vector a. We often write aR2 to denote that it can be described by two real coordinates.

the magnitude or the length formula of a vector is:

For two point in Cartesian space coordinate so, the magnitude of a vector is:

We saw earlier how to represent 2-dimensional vectors on the xy plane. Now we extend the idea to represent 3-dimensional vectors using the xyz axes.

If so the distance between A and B is:

And Vectors are vectors of length 1 in the directions respectively.

Operations on vectors

  • Addition of vectors

Given two vectors https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1  and https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0062.png?rev=2.6.1  , we form their sum https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0062.png?rev=2.6.1  , as follows. We translate the vector https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0062.png?rev=2.6.1   until its tail coincides with the head of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1  . (Recall such translation does not change a vector.) Then, the directed line segment from the tail off https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   to the head of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0062.png?rev=2.6.1   is the vector https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/002B.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0062.png?rev=2.6.1  .

Addition of vectors satisfies two important properties.

.

  • Vector subtraction

We define subtraction as addition with the opposite of a vector:

  • Scalar multiplication

Given a vector https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   and a real number (scalar) https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1  , we can form the vector https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   as follows. If https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1   is positive, then https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   is the vector whose direction is the same as the direction of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   and whose length is https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1   times the length of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1  . In this case, multiplication by https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1   simply stretches (if https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003E.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png?rev=2.6.1  ) or compresses (if https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0030.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003C.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003C.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png?rev=2.6.1  ) the vector https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1

.If, on the other hand, https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1  is negative, then we have to take the opposite of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   before stretching or compressing it. In other words, the vector https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   points in the opposite direction of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1  , and the length of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   is https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/007C.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/007C.png?rev=2.6.1   times the length of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1  . No matter the sign of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1  , we observe that the magnitude of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1   is https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/007C.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/007C.png?rev=2.6.1   times the magnitude of https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1  : https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/003D.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/007C.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Math/Italic/120/03BB.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/007C.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Bold/120/0061.png?rev=2.6.1 https://mathinsight.org/static/MathJax/fonts/HTML-CSS/TeX/png/Main/Regular/120/2225.png?rev=2.6.1

Scalar multiplications satisfy many of the same properties as the usual multiplication.

  • Dot product

the Dot Product gives a number as an answer (a “scalar”, not a vector). The Dot Product is written using a central dot This means the Dot Product of a and b.

We can calculate the Dot Product of two vectors this way

  • Cross product

Two vectors can be multiplied using the “Cross Product”. The Cross Product a × b of two vectors is another vector that is at right angles to both:

We can calculate the Cross Product this way:

n is the unit vector at right angles to both a and b.

So, the length is: the length of a times the length of b times the sine of the angle between a and b,

Then we multiply by the vector n to make sure it heads in the right direction (at right angles to both a and b).

The cross product could point in the completely opposite direction and still be at right angles to the two other vectors, so we have the: “Right Hand Rule”

           3. Summary:

In this course, the student knows about the vectors and how they are defined in the cartesian coordinates and have a good idea in the most important characteristics of the operations on the vectors.

           4. References:

[1] James Stewart, Calculus (Early Trancendentals) 3rd Ed., Brooks/Cole, Pacific Grove, CA, 1995.

 

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