Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 709

ESSENTIAL FORMULAE 709

Complex equations:Ifm+jn=p+jqthenm=p

andn=q

Multiplication:z 1 z 2 =r 1 r 2 ∠(θ 1 +θ 2 )

Division:

z 1

z 2

=

r 1

r 2

∠(θ 1 −θ 2 )

De Moivre’s theorem:

[r∠θ]n=rn∠nθ=rn(cosnθ+jsinnθ)=rejθ

Matrices and Determinants

Matrices:

IfA=

(

ab

cd

)

and B=

(

ef

gh

)

then

A+B=

(

a+eb+f

c+gd+h

)

A−B=

(

a−eb−f

c−gd−h

)

A×B=

(

ae+bg af+bh

ce+dg cf+dh

)

A−^1 =

1

ad−bc

(

d −b

−ca

)

If A=

(

a 1 b 1 c 1

a 2 b 2 c 2

a 3 b 3 c 3

)

then A−^1 =

BT

|A|

where

BT=transpose of cofactors of matrix A

Determinants:

∣

∣

∣

∣

ab

cd

∣

∣

∣

∣=ad−bc

∣

∣

∣

∣

∣

a 1 b 1 c 1

a 2 b 2 c 2

a 3 b 3 c 3

∣

∣

∣

∣

∣

=a 1

∣

∣

∣

∣

b 2 c 2

b 3 c 3

∣

∣

∣

∣−b^1

∣

∣

∣

∣

a 2 c 2

a 3 c 3

∣

∣

∣

∣

+c 1

∣

∣

∣

∣

a 2 b 2

a 3 b 3

∣

∣

∣

∣

Differential Calculus

Standard derivatives

yorf(x)

dy

dx

orf′(x)

axn anxn−^1

sinax acosax

cosax −asinax

tanax asec^2 ax

secax asecaxtanax

cosecax −acosecaxcotax

cotax −acosec^2 ax

eax aeax

lnax

1

x

sinhax acoshax

coshax asinhax

tanhax asech^2 ax

sechax −asechaxtanhax

cosechax −acosechaxcothax

cothax −acosech^2 ax

sin−^1

x

a

1

√

a^2 −x^2

sin−^1 f(x)

f′(x)

√

1 −[f(x)]^2

cos−^1

x

a

− 1

√

a^2 −x^2

cos−^1 f(x)

−f′(x)

√

1 −[f(x)]^2

tan−^1

x

a

a

a^2 +x^2

tan−^1 f(x)

f′(x)

1 +[f(x)]^2

sec−^1

x

a

a

x

√

x^2 −a^2

sec−^1 f(x)

f′(x)

f(x)

√

[f(x)]^2 − 1

cosec−^1

x

a

−a

x

√

x^2 −a^2