8/2/2019 HMS112FormulaSheets_sem2

1/5

8/2/2019 HMS112FormulaSheets_sem2

2/5

HMS 112 FORMULA SHEETS 2

(i) the surface has a local maximum at P if

2z

x2< 0

and

2z

x22z

y2(

2z

xy)2 > 0

;

(ii) the surface has a local minimum at P if

2z

x2> 0

and

2z

x22z

y2(

2z

xy)2 > 0

;

(iii) the surface has a saddle point at P if

2z

x22z

y2(

2z

xy)2 < 0

CHAIN RULES:

(i) ifz = f(x,y) and x = x(t) and y = y(t) thendz

dt=z

x

dx

dt+z

y

dy

dt

(ii) ifz = f(x,t) and x = x(t) then

dz

dt =

z

x

dx

dt +

z

t

(iii) if z = f(x,y) and y = y(x) then

dz

dx=z

x+z

y

dy

dx

(iv) if z = f(x,y) and x = x(u,v) and y = y(u,v) thenz

u=z

x

x

u+z

y

y

u and

z

v=z

x

x

v+z

y

y

v

COMPLEX IMPEDANCE IN RLC CIRCUITS

= 2f

Z

VI

cisZZ

CLjRZ

tII

tVV

00

0

0

)1

(

)sin(

)sin(

=

=

+=

=

+=

LINEAR HOMOGENEOUS 2ND ORDER DIFFERENTIAL EQUATIONS WITHCONSTANT COEFFICIENTS

i.e.ad2y

dx2+ b

dy

dx+ cy = 0

(1)

IF THE ROOTS OF THECHARACTERISTIC (AUXILIARY)

EQUATION am2 + bm + c = 0 ARE

THEN THE SOLUTIONOF (1) IS

2 real and different roots m1 and m2 y = Aem1x + Be

m2x

8/2/2019 HMS112FormulaSheets_sem2

3/5

HMS 112 FORMULA SHEETS 3

2 real and equal roots ( i.e. m = m1 = m2 )(also called a double root)

y = em1x A +Bx( )

2 conjugate complex roots m1 and m2 ,

where m1 = + j and m2 = j

(where1=

j )

y = ex

(A cosx + Bsinx)

LINEAR NONHOMOGENEOUS 2ND ORDER DIFFERENTIAL EQUATIONSWITH CONSTANT COEFFICIENTS

i.e.ad2y

dx2+ b

dy

dx+ cy = f(x)

, where f(x) 0 (2)

The solution of (2) is phyyy +=

, where)( ch yy (the COMPLEMENTARY

FUNCTION or C.F.) is the GENERAL SOLUTION (i.e. it contains 2 arbitraryconstants) of the corresponding homogeneous equation, i.e.

ad2y

dx2+ b

dy

dx+ cy = 0

andyp , the PARTICULAR INTEGRAL, can be found by the table below:

IF f(x) HAS A TERM ACONSTANTMULTIPLE OF

AND IF THENy

p IS

epx p N.R.A.E (i.e. p not a

root of auxiliaryequation)

Cepx

p S.R.A.E (i.e. p a singleroot of the auxiliaryequation)

Cxepx

p D.R.A.E (i.e. p adouble root of theauxiliaryequation)

Cx2epx

sinqx orcosqx jq N.R.A.E (where1=j )

Ccosqx + Dsin qx

jq S.R.A.E x(Ccos qx + Dsin qx)

a0 + a1x + .... + anxn 0 N.R.A.E. b0 + b1x + .... + bnx n

0 S.R.A.E. x(b0 + b1x + .... + bnxn)

0 D.R.A.E. x2(b0 + b1x + .... + bnx

n)

Note: the constants inyp can always be found.

Derivatives

8/2/2019 HMS112FormulaSheets_sem2

4/5

HMS 112 FORMULA SHEETS 4

FUNCTION DERIVATIVE

sin1 x

1

1 x2

cos1 x 21

1

x

tan1 x

1

1+ x2

sinh1x

1

1 + x2

cosh1x

1

x2

1

tanh1x

1

1 x2

Integrals

FUNCTION INTEGRAL1

a2

x2

ca

x+

1sin

1

x2 + a2c

a

x

a+

1tan1

sinh kx cosh kx

k+ c

cosh kx sinhkx

k + csech

2kx tanh kx

k+ c

1

a2

+ x2

sinh1 x

a+ c

orln( x + x2 + a2 )+ c

1

x2

a2

cosh1 x

a+ c

or caxx ++ )ln(22

1

a2 x21

atanh

1 x

a+ c

orc

xa

xa

a+

+ln

2

1

1

x2 a2 cax

a+

1coth

1

orc

ax

ax

a ++

ln2

1

xe x sin xex cos

c

xxe x

+

+

)cossin(22

xe x cos

)sincos(22

xxe x

++

+c

INTEGRATION BY PARTS: u

dv

dx dx = uv vdu

dx dx

8/2/2019 HMS112FormulaSheets_sem2

5/5

HMS 112 FORMULA SHEETS 5

SUBSTITUTION RULE:f(u)

du

dxdx = f(u) du

orf(x)dx = f(x)

dx

d d

RULES OF BOOLEAN ALGEBRA:

p + p = p (1)pp = p (2)p + 0 = p (3)p +1= 1 (4)p1=p (5)p0 = 0 (6)p +p =1 (7)pp = 0 (8)

p =p (9)p + qr= (p + q)(p + r) (10)p + pr= p (11)