Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 708

708 ESSENTIAL FORMULAE

s

θ

r

r

Figure FA3

For sector of circle:

s=rθ (θin rad)

shaded area=^12 r^2 θ (θin rad)

Equation of a circle, centre at (a, b), radius r:

(x−a)^2 +(y−b)^2 =r^2

Linear and angular velocity

If v=linear velocity (m/s), s=displacement (m),

t=time (s),n=speed of revolution (rev/s),

θ=angle (rad),ω=angular velocity (rad/s),

r=radius of circle (m) then:

v=

s

t

ω=

θ

t

= 2 πn v=ωr

centripetal force=

mv^2

r

wherem=mass of rotating object.

Graphs

Equations of functions

Equation of a straight line: y=mx+c

Equation of a parabola: y=ax^2 +bx+c

Circle, centre (a, b), radius r:

(x−a)^2 +(y−b)^2 =r^2

Equation of an ellipse, centre at origin, semi-axes

a and b:

x^2

a^2

+

y^2

b^2

= 1

Equation of a hyperbola:

x^2

a^2

−

y^2

b^2

= 1

Equation of a rectangular hyperbola: xy=c^2

Irregular areas

Trapezoidal rule

Area≈

(

width of

interval

)[

1

2

(

first+last

ordinates

)

+

(

sum of remaining

ordinates

)]

Mid-ordinate rule

Area≈

(

width of

interval

)(

sum of

mid-ordinates

)

Simpson’s rule

Area≈

1

3

(

width of

interval

)[(

first+last

ordinate

)

+ 4

(

sum of even

ordinates

)

+ 2

(

sum of remaining

odd ordinates

)]

Vector Geometry

Ifa=a 1 i+a 2 j+a 3 kandb=b 1 i+b 2 j+b 3 k

a·b=a 1 b 1 +a 2 b 2 +a 3 b 3

|a|=

√

a^21 +a^22 +a^23 cosθ=

a·b

|a||b|

a×b=

∣

∣

∣

∣

∣

ijk

a 1 a 2 a 3

b 1 b 2 b 3

∣

∣

∣

∣

∣

|a×b|=

√

[(a·a)(b·b)−(a·b)^2 ]

Complex Numbers

z=a+jb=r(cosθ+jsinθ)=r∠θ=rejθ where

j^2 =− 1

Modulusr=|z|=

√

(a^2 +b^2 )

Argumentθ=argz=tan−^1

b

a

Addition:(a+jb)+(c+jd)=(a+c)+j(b+d)

Subtraction:(a+jb)−(c+jd)=(a−c)+j(b−d)