Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 2 ppt

non-Chapter 2Vectors 2.1 Vectors 2.1.1 Scalars and Vectors A vector is a quantity having both a magnitude and a direction.. Two vectors are equal if they have the same magnitude and dire

1 2 1

2

1 2

Hint 1.8

Write the polynomial in factored form

1.7 Solutions

Solution 1.1

area = π × radius2area = π

4 × diameter2The constant of proportionality is π4

We note that one or both sides of the equation are undefined at y = ±2 because of division by zero There are

no solutions for these two values of y and we assume from this point that y 6= ±2 We multiply by (y − 2)(y + 2)

(x + 1)(y + 2) = (x + 1)(x − 1)For x = −1, the equation becomes the identity 0 = 0 Now we consider x 6= −1 We divide by x + 1 to obtainthe equation of a line

y + 2 = x − 1

y = x − 3Now we collect the solutions we have found

{(−1, y) : y 6= ±2} ∪ {(x, x − 3) : x 6= 1, 5}

The solutions are depicted in Figure /reffig not a line

-6 -4 -2 2 4 6

-6 -4 -2

2 4

Figure 1.12: The solutions of x+1y−2 = xy22−1−4

= 10

189

11180 ≈ 35.4

Figure 1.13: Graphs of f (−x), f (x + 3), f (3 − x) + 2, and f−1(x).

Note that p(x) and q(x) appear as a ratio, they are determined only up to a multiplicative constant We may takethe leading coefficient of q(x) to be unity.

= lim

x→∞

2a2

= a

2

x2+ βx + χNow we use the fact that f (x) is even to conclude that q(x) is even and thus β = 0

f (x) = 2x

2

x2+ χFinally, we use that f (1) = 1 to determine χ

f (x) = 2x

2

x2+ 1

non-Chapter 2

Vectors

2.1 Vectors

2.1.1 Scalars and Vectors

A vector is a quantity having both a magnitude and a direction Examples of vector quantities are velocity, forceand position One can represent a vector in n-dimensional space with an arrow whose initial point is at the origin,(Figure 2.1) The magnitude is the length of the vector Typographically, variables representing vectors are oftenwritten in capital letters, bold face or with a vector over-line, A, a, ~a The magnitude of a vector is denoted |a|

A scalar has only a magnitude Examples of scalar quantities are mass, time and speed

Vector Algebra Two vectors are equal if they have the same magnitude and direction The negative of a vector,denoted −a, is a vector of the same magnitude as a but in the opposite direction We add two vectors a and b byplacing the tail of b at the head of a and defining a + b to be the vector with tail at the origin and head at the head

of b (See Figure2.2.)

The difference, a − b, is defined as the sum of a and the negative of b, a + (−b) The result of multiplying a by

a scalar α is a vector of magnitude |α| |a| with the same/opposite direction if α is positive/negative (See Figure 2.2.)

-a

a

2a

Figure 2.2: Vector arithmetic

Here are the properties of adding vectors and multiplying them by a scalar They are evident from geometric

(a + b) + c = a + (b + c) α(βa) = (αβ)a associative lawsα(a + b) = αa + αb (α + β)a = αa + βa distributive laws

Zero and Unit Vectors The additive identity element for vectors is the zero vector or null vector This is a vector

of magnitude zero which is denoted as 0 A unit vector is a vector of magnitude one If a is nonzero then a/|a| is aunit vector in the direction of a Unit vectors are often denoted with a caret over-line, ˆn

Rectangular Unit Vectors In n dimensional Cartesian space, Rn, the unit vectors in the directions of thecoordinates axes are e1, en These are called the rectangular unit vectors To cut down on subscripts, the unitvectors in three dimensional space are often denoted with i, j and k (Figure 2.3)

x

z

y

j k

i

Figure 2.3: Rectangular unit vectors

Components of a Vector Consider a vector a with tail at the origin and head having the Cartesian coordinates(a1, , an) We can represent this vector as the sum of n rectangular component vectors, a = a1e1 + · · · + anen.(See Figure 2.4.) Another notation for the vector a is ha1, , ani By the Pythagorean theorem, the magnitude ofthe vector a is |a| =pa2

Figure 2.4: Components of a vector

2.1.2 The Kronecker Delta and Einstein Summation Convention

The Kronecker Delta tensor is defined

δij =

(

1 if i = j,

0 if i 6= j

This notation will be useful in our work with vectors

Consider writing a vector in terms of its rectangular components Instead of using ellipses: a = a1e1+ · · · + anen, wecould write the expression as a sum: a =Pn

i=1aiei We can shorten this notation by leaving out the sum: a = aiei,where it is understood that whenever an index is repeated in a term we sum over that index from 1 to n This is the

Einstein summation convention A repeated index is called a summation index or a dummy index Other indices cantake any value from 1 to n and are called free indices.

Example 2.1.1 Consider the matrix equation: A · x = b We can write out the matrix and vectors explicitly

2.1.3 The Dot and Cross Product

Dot Product The dot product or scalar product of two vectors is defined,

a · b ≡ |a||b| cos θ,where θ is the angle from a to b From this definition one can derive the following properties:

• a · b = b · a, commutative

• α(a · b) = (αa) · b = a · (αb), associativity of scalar multiplication

• a · (b + c) = a · b + a · c, distributive (See Exercise 2.1.)

• eiej = δij In three dimensions, this is

i · i = j · j = k · k = 1, i · j = j · k = k · i = 0

• a · b = aibi ≡ a1b1+ · · · + anbn, dot product in terms of rectangular components

• If a · b = 0 then either a and b are orthogonal, (perpendicular), or one of a and b are zero

The Angle Between Two Vectors We can use the dot product to find the angle between two vectors, a and

b From the definition of the dot product,

√2

x · n = a · n

The normal to a line determines an orientation of the line The normal points in the direction that is above theline A point b is (above/on/below) the line if (b − a) · n is (positive/zero/negative) The signed distance of a point

Figure 2.5: Equation for a line.

b from the line x · n = a · n is

(b − a) · n

|n|.

Implicit Equation of a Hyperplane A hyperplane in Rn is an n − 1 dimensional “sheet” which passes through

a given point and is normal to a given direction In R3 we call this a plane Consider a hyperplane that passes throughthe point a and is normal to the vector n All the hyperplanes that are normal to n have the property that x · n is aconstant, where x is any point in the hyperplane x · n = 0 is the hyperplane that is normal to n and passes throughthe origin The hyperplane that is normal to n and passes through the point a is

x · n = a · n

The normal determines an orientation of the hyperplane The normal points in the direction that is above thehyperplane A point b is (above/on/below) the hyperplane if (b − a) · n is (positive/zero/negative) The signed

distance of a point b from the hyperplane x · n = a · n is

(b − a) · n

|n|.

Right and Left-Handed Coordinate Systems Consider a rectangular coordinate system in two dimensions.Angles are measured from the positive x axis in the direction of the positive y axis There are two ways of labeling theaxes (See Figure 2.6.) In one the angle increases in the counterclockwise direction and in the other the angle increases

in the clockwise direction The former is the familiar Cartesian coordinate system

x y

Figure 2.6: There are two ways of labeling the axes in two dimensions

There are also two ways of labeling the axes in a three-dimensional rectangular coordinate system These are calledright-handed and left-handed coordinate systems See Figure 2.7 Any other labelling of the axes could be rotated intoone of these configurations The right-handed system is the one that is used by default If you put your right thumb inthe direction of the z axis in a right-handed coordinate system, then your fingers curl in the direction from the x axis

to the y axis

Cross Product The cross product or vector product is defined,

a × b = |a||b| sin θ n,where θ is the angle from a to b and n is a unit vector that is orthogonal to a and b and in the direction such thatthe ordered triple of vectors a, b and n form a right-handed system

z

y

j i

Figure 2.7: Right and left handed coordinate systems

You can visualize the direction of a × b by applying the right hand rule Curl the fingers of your right hand in thedirection from a to b Your thumb points in the direction of a × b Warning: Unless you are a lefty, get in the habit

of putting down your pencil before applying the right hand rule

The dot and cross products behave a little differently First note that unlike the dot product, the cross product is notcommutative The magnitudes of a × b and b × a are the same, but their directions are opposite (See Figure2.8.)Let

a × b = |a||b| sin θ n and b × a = |b||a| sin φ m

The angle from a to b is the same as the angle from b to a Since {a, b, n} and {b, a, m} are right-handed systems,

m points in the opposite direction as n Since a × b = −b × a we say that the cross product is anti-commutative

Next we note that since

|a × b| = |a||b| sin θ,the magnitude of a × b is the area of the parallelogram defined by the two vectors (See Figure 2.9.) The area of thetriangle defined by two vectors is then 12|a × b|

From the definition of the cross product, one can derive the following properties:

a b

• i × j = k, j × k = i, k × i = j.

•

a × b = (a2b3− a3b2)i + (a3b1 − a1b3)j + (a1b2− a2b1)k =

a1 a2 a3

b1 b2 b3

Then note that that dot product of any two distinct rectangular unit vectors is zero because they are orthogonal Now

we write a and b in terms of their rectangular components and use the distributive law

a · b = aiei· bjej

= aibjei· ej

= aibjδij

= aibiSolution 2.3

Since a · b = |a||b| cos θ, we have

θ = arccos a · b

|a||b|



Figure 2.12: The distributive law for the dot product.

when a and b are nonzero

√

2√10



= arccos 2

√55

!

≈ 0.463648

Solution 2.4

First consider the case that both b and c are orthogonal to a b + c is the diagonal of the parallelogram defined by

b and c, (see Figure 2.13) Since a is orthogonal to each of these vectors, taking the cross product of a with thesevectors has the effect of rotating the vectors through π/2 radians about a and multiplying their length by |a| Note

that a × (b + c) is the diagonal of the parallelogram defined by a × b and a × c Thus we see that the distributive lawholds when a is orthogonal to both b and c,

a × (b + c) = a × b + a × c

b

c b+c

a c a

a b

a (b+c)

Figure 2.13: The distributive law for the cross product

Now consider two arbitrary vectors a and b We can write b = b⊥+ bk where b⊥ is orthogonal to a and bk isparallel to a, (see Figure2.14)

By the definition of the cross product,

a × b = |a||b| sin θ n

Note that

|b⊥| = |b| sin θ,

Solution 2.5

We know that

i × j = k, j × k = i, k × i = j,and that

a1 a2 a3

b1 b2 b3

= i

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Figure 2 . 12 : The distributive law for the dot product.

when a and b are nonzero

√

2? ? ?10



= arccos 2< /sup>

√55

!...

Figure 2 .11 : Three points define a plane

2. 2 Sets of Vectors in n Dimensions

Orthogonality Consider two n-dimensional vectors

x = (x1< /small>, x2< /small>,... data-page=" 32" >

Solution 2. 5

We know that

i × j = k, j × k = i, k × i = j ,and that

a1< /sub> a2< /sub> a3

b1< /sub> b2< /sub> b3