Handbook of mathematics for engineers and scienteists part 17 pps

Suppose that two rectangular Cartesian coordinate systems OXY and $ O $ X $ Y are given and the first system is taken to the second by the translation of the origin O0,0 of the first sys

O O O

A

x

x

x

x y

y

y

x X

α

α

X X

X

X

X

Y Y

Figure 4.3 Transformation of Cartesian coordinates under parallel translation (a), under rotation (b), and

under translation and rotation (c) of axes.

Let an arbitrary point A have coordinates (x, y) in the system OXY and coordi-nates ( ˆx, ˆy) in the system O $ X $ Y The transformation of rectangular Cartesian coordinates

by the rotation of axes is given by the formulas

x = ˆx cos α – ˆy sin α,

y = ˆx sin α + ˆy cos α or

ˆx = x cos α + y sin α,

ˆy = –x sin α + y cos α. (4.1.2.3)

4.1.2-4 Transformation of coordinates under translation and rotation of axes

Suppose that two rectangular Cartesian coordinate systems OXY and $ O $ X $ Y are given and

the first system is taken to the second by the translation of the origin O(0,0) of the first system to the origin $O (x0, y0) of the second system followed by the rotation of the system around the point $O by an angle α (see Fig 4.3c and Paragraphs 4.1.2-2 and 4.1.2-3) Let an arbitrary point A have coordinates (x, y) in the system OXY and coordi-nates ( ˆx, ˆy) in the system $ O $ X $ Y The transformation of rectangular Cartesian coordinates

by the parallel translation and rotation of axes is given by the formulas

x = ˆx cos α – ˆy sin α + x0,

y = ˆx sin α + ˆy cos α + y0, or

ˆx = (x – x0) cos α + (y – y0) sin α,

ˆy = –(x – x0) sin α + (y – y0) cos α. (4.1.2.4)

4.1.2-5 Polar coordinates

A polar coordinate system is determined by a point O called the pole, a ray OA issuing from this point, which is called the polar axis, a scale segment for measuring lengths, and

the positive sense of rotation around the pole Usually, the anticlockwise sense is assumed

to be positive (see Fig 4.4a).

The position of each point B on the plane is determined by two polar coordinates, the

polar radius ρ =|OB|and the polar angle θ = ∠AOB (the values of the angle θ are defined

up to the addition of 2πn , where n is an integer) To be definite, one usually assumes that

0 ≤θ≤ 2π or –π≤θ≤π The polar radius of the pole is zero, and its polar angle does not have any definite value

4.1.2-6 Relationship between Cartesian and polar coordinates

Suppose that B is an arbitrary point on the plane, (x, y) are its rectangular Cartesian coor-dinates, and (ρ, θ) are its polar coordinates (see Fig 4.4b) The formulas of transformation

O A O

B B

y

Y

Figure 4.4 A polar coordinate system (a) Relationship between Cartesian and polar coordinates (b).

from one coordinate system to the other have the form

x = ρ cos θ,

ρ=

x2+ y2,

where the polar angle θ is determined with regard to the quadrant where the point B lies.

Example 2 Let us find the polar coordinates ρ, θ (0 ≤ θ≤ 2π ) of the point B whose Cartesian coordinates are x = –3, y = –3

From formulas (4.1.2.5), we obtain ρ =

(– 3 )2+ (– 3 )2= 3√2and tan θ = –3

– 3 =1 Since the point B lies

in the third quadrant, we have θ = arctan1+ π = 5

4π.

4.1.3 Points and Segments on Plane

4.1.3-1 Distance between points on plane

The distance d between two arbitrary points A1and A2on the plane is given by the formula

d=

(x2– x1)2+ (y2– y1)2, (4.1.3.1)

where x and y with the corresponding subscripts are the Cartesian coordinates of these

points, and by the formula

d=



ρ2

1+ ρ22–2ρ1ρ2cos(θ2– θ1), (4.1.3.2)

where ρ and θ with the corresponding subscripts are the polar coordinates of these points.

4.1.3-2 Segment and its projections

Suppose that an axis u and an arbitrary segment −−→ A1A2are given on the plane (see Fig 4.5a) From the points A1 and A2, we draw the perpendiculars to u and denote the points of intersection of the perpendiculars with the axis by P1 and P2 The value P1P2 of the

segment −−→ P1P2of the axis u is called the projection of the segment −−→ A1A2onto the axis u. Usually one writes pru A −−→

1A2= P1P2 If ϕ (0 ≤ϕ≤π ) is the angle between the segment −−→ A1A2

and the axis u, then

pru −−→ A

For two arbitrary points A1(x1, y1) and A2(x2, y2), the projections x and y of the segment −−→ A1A2onto the coordinate X- and Y-axes are given by the formulas (see Fig 4.5b)

prX A −−→

1A2 = x2– x1, prY A −−→

1A2= y2– y1. (4.1.3.4)

A

A

A A

d θ φ

x y

y

2 2

1 1

1 2

Y

Figure 4.5 Projection of the segment onto the axis u (a) and onto the coordinate X- and Y-axes (b).

Thus, to obtain the projections of a segment onto the coordinate axes, one subtracts the coordinates of its initial point from the respective coordinates of its endpoint

The projections of the segment −−→ A1A2onto the coordinate axes can be found if its length d (see (4.1.3.1)) and polar angle θ are known (see Fig 4.5b) The corresponding formulas

are

prX A −−→

1A2= d cos θ, prY A −−→

1A2= d sin θ, tan θ = x y2– y1

2– x1 . (4.1.3.5) 4.1.3-3 Angles between coordinate axes and segments

The angles α x≡θ and α y between the segment −−→ A1A2and the coordinate x- and y-axes are determined by the expressions

cos α x = x2– x1

(x2– x1)2+ (y2– y1)2 , cos α y =

y2– y1

(x2– x1)2+ (y2– y1)2 , (4.1.3.6)

and α y = π – α x

The angle β between arbitrary segments −−→ A1A2and −−→ A3A4joining the points A1(x1, y1),

A2(x2, y2) and A3(x3, y3), A4(x4, y4), respectively, can be found from the relation

cos β = (x2– x1)(x4– x3) + (y2– y1)(y4– y3)

(x2– x1)2+ (y2– y1)2

(x4– x3)2+ (y4– y3)2 . (4.1.3.7)

4.1.3-4 Division of segment in given ratio

The number λ = p/q, where p = A1A and q = AA2are the values of the directed segments −−→ A1A

and −→ AA2, is called the ratio in which point A divides the segment −−→ A1A2 It is independent

of the sense of the segment (i.e., one could use the segment −−→ A2A1) and the scale segment.

The coordinates of the point A dividing the segment −−→ A1A2 in a ratio λ are given by the formulas

x= x1+ λx2

1+ λ =

qx1+ px2

q + p , y=

y1+ λy2

1+ λ =

qy1+ py2

q + p , (4.1.3.8) where –∞≤λ≤∞.

For the coordinates of the midpoint of the segment −−→ A1A2, we have

x= x1+ x2

2 , y=

y1+ y2

2 ; (4.1.3.9)

i.e., each coordinate of the midpoint of a segment is equal to the half-sum of the respective coordinates of its endpoints

4.1.3-5 Area of triangle area.

The area S3of the triangle with vertices A1, A2, and A3is given by the formula

S3 = 1

2[(x2– x1)(y3– y1) – (x3– x1)(y2– y1)]

= 1

2 x2– x1 y2– y1

x3– x1 y3– y1



 = 12 





x1 y1 1

x2 y2 1

x3 y3 1





, (4.1.3.10)

where x and y with respective subscripts are the Cartesian coordinates of the vertices, and

by the formula

S3= 1

2[ρ1ρ2sin(θ2– θ1) + ρ2ρ3sin(θ3– θ2) + ρ3ρ1sin(θ1– θ3)], (4.1.3.11)

where ρ and θ with respective subscripts are the polar coordinates of the vertices In

formulas (4.1.3.10) and (4.1.3.11), one takes the plus sign if the vertices are numbered

anticlockwise (see Fig 4.6a) and the minus sign otherwise.

A A

A

A A

A

2 3

n-1 n

2

1 1

3

Figure 4.6 Area of triangle (a) and of a polygon (b).

4.1.3-6 Area of a polygon

The area S n of the polygon with vertices A1, , A nis given by the formula

S n= 1

2[(x1– x2)(y1+ y2) + (x2– x3)(y2+ y3) +· · · + (x n – x1)(y n + y1)], (4.1.3.12)

where x and y with respective subscripts are the Cartesian coordinates of the vertices, and

by the formula

S n= 1

2[ρ1ρ2sin(θ2– θ1) + ρ2ρ3sin(θ3– θ2) +· · · + ρ n ρ1sin(θ1– θ n)], (4.1.3.13)

where ρ and θ with respective subscripts are the polar coordinates of the vertices In

formulas (4.1.3.12) and (4.1.3.13), one takes the plus sign if the vertices are numbered

anticlockwise (see Fig 4.6b) and the minus sign otherwise.

Remark. One often says that formulas (4.1.3.10)–(4.1.3.13) express the oriented area of the corresponding

figures.

4.2 Curves on Plane

4.2.1 Curves and Their Equations

4.2.1-1 Basic definitions

A curve on the plane determined by an equation in some coordinate system is the geometric

locus of points of the plane whose coordinates satisfy this equation

An equation of a curve on the plane in a given coordinate system is an equation with

two variables such that the coordinates of the points lying on the curve satisfy the equation and the coordinates of the points that do not lie on the curve do not satisfy it

The coordinates of an arbitrary point of a curve occurring in an equation of the curve

are called current coordinates.

4.2.1-2 Equation of curve in Cartesian coordinate system

An equation of a curve in the Cartesian coordinate system OXY can be written as

The image of a curve determined by an equation of the form

is called the graph of the function f (x).

Example 1 Let us plot the curve determined by the equation x2– y =0 We express one coordinate via

the other (e.g., y via x) from this equation: y = x2 Specifying various values of x, we find the corresponding values of y and thus construct a sequence of points of the desired curve By joining these points, we obtain the curve itself (see Fig 4.7a).

4

8

12

16

O

5

X

x

4 3 1 1 3

2 0 2 4

y

16 9 1 1 9

4 0 4

X

Y

Y

y = x

y= x

2

Y

Figure 4.7 Cartesian coordinate system Loci of points for equations x2 – y =0(a), y –5 = 0(b), and

x2– y2= 0(c).

4.2.1-3 Special kinds of equations

1 The equation of a curve on the plane may contain only one of the current coordinates but still determine a certain curve

Example 2 Suppose that the equation y –5 = 0(or y =5 ) is given The locus of points whose coordinates

are equal to five is the straight line parallel to the axis OX and passing through the point y =5of the axis OY (see Fig 4.7b).

Similarly, the equation x +7 = 0determines a straight line parallel to the axis OY

2 If the left-hand side of equation (4.2.1.1) can be factorized, then, equating each factor separately with zero, we obtain several new equations, each of which can determine a certain curve

Example 3 Consider the equation x2 – y2 = 0 Factorizing the left-hand side of this equation, we

obtain (x + y)(x – y) =0 Obviously, the latter equation determines the pair of straight lines x + y =0

and x – y =0, which are the bisectors of the coordinate angles (see Fig 4.7c).

3 Equation (4.2.1.1) may determine a locus consisting of one or several isolated points

Example 4 The equation x2+ y2= 0 determines the single point with coordinates ( 0 , 0 ).

Example 5 The equation (x2– 9 )2+ (y2– 25 )2 = 0 defines the locus consisting of the four points ( 3 , 5 ), ( 3 , – 5 ), (– 3 , 5 ), and (– 3 , – 5 ).

4 There exist equations that do not determine any locus

Example 6 The equation x2+ y2+ 5 = 0does not have solutions for any real x and y.

4.2.1-4 Equation of curve in polar coordinate system

An equation of a curve in a polar coordinate system can be written as

where ρ is the polar radius and θ is the polar angle This equation is satisfied by the polar

coordinates of any point lying on the curve and is not satisfied by the coordinates of the points that do not lie on the curve

Example 7 Consider the equation ρ – a cos θ =0(or ρ = a cos θ), where a is a positive number By B

we denote the point with polar coordinates (ρ, θ), and by A we denote the point with coordinates (a,0 ).

If ρ = a cos θ, then the angle OBA is a right angle, and vice versa Therefore, the locus of points whose coordinates satisfy this equation is a circle with diameter a (see Fig 4.8).

a ρ

B

Figure 4.8 Polar coordinate system Locus of points for equations ρ – a cos θ =0

Example 8 Consider the equation ρ – aθ =0(or ρ = aθ), where a is a positive constant The curve determined by this equation is called a spiral of Archimedes.

As θ increases starting from zero, the point B(ρ, θ) issues from the pole and moves around it in the positive sense, simultaneously moving away from it For each point of this curve with positive coordinates (ρ, θ), one has the corresponding point (–ρ, –θ) on the same curve Figures 4.9a and b show the branches of the spiral of Archimedes corresponding to the positive and negative values of θ, respectively.

Note that the spiral of Archimedes divides each polar ray into equal segments (except for the segment nearest to the pole).

O

B

B

( )a

θ > 0

2aπ 2aπ 2aπ

( )b

θ < 0

Figure 4.9 A spiral of Archimedes ρ = aθ corresponding to the positive (a) and negative (b) values of θ.

O

O a

B

( )a

θ < 0

θ > 0

ρ θ

( )b

Figure 4.10 A hyperbolic spiral ρ = a/θ corresponding to the positive (a) and negative (b) values of θ.

( )a

( )b

Figure 4.11 A logarithmic spiral ρ = a θ corresponding to the values a >1(a) and0< a <1(b).

Example 9 Consider the equation ρ – a/θ =0(or ρ = a/θ), where a is a positive number The curve determined by this equation is called a hyperbolic spiral.

As θ increases, the point B(ρ, θ) moves around the pole in the positive sense while approaching it endlessly.

As θ tends to zero, the point B approaches the line y = a while moving to infinity Figures 4.10a and b show the branches of the hyperbolic spiral corresponding to the positive and negative values of θ, respectively.

Example 10 Consider the equation ρ – a θ = 0(or ρ = a θ ), where a is a positive number The curve determined by this equation is called the logarithmic spiral.

Figures 4.11a and b show the branches of the logarithmic spiral corresponding to the values a > 1

and 0< a <1, respectively For a =1 , we obtain the equation of a circle.