Mathematics Study Material

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM/IISc, JEST, TIFR and GRE in PHYSICAL SCIENCES

Basic Mathematics Formula Sheet for Physical Sciences

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Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 1

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

Basic Mathematics Formula Sheet for Physical Sciences

1. Trigonometry…………………………………………………………………… (3-9) 1.1 Trigonometrical Ratios and Identities………………………………………...… (3-7) 1.2 Inverse Circular Functions………………………………………...……………. (8-9)

2. Differential and integral Calculus………………………………………….… (10-20) 2.1 Differentiation …………...……………. ……………………………………... (10-12) 2.2 Limits …………………………………………………………………..………(13-14) 2.3 Tangents and Normal……………………………..…………………………….(15-16) 2.4 Maxima and Minima ……………………………………………………………..(16) 2.5 Integration……………………………………………………………………....(17-19) 2.5.1 Gamma integral……………………………………………………..(19)

3. Differential Equations………………………………………………….……....(20-22)

4. Vectors……………….…….................................................................................(23-25)

5. Algebra……………….…….................................................................................(26-32) 5.1 Theory of Quadratic equations…………………………………………………...(26) 5.2 Logarithms………………………………………………………………………..(27) 5.3 Permutations and Combinations……………………………………………......(28-29) 5.4 Binomial Theorem…………………………………………………......................(30) 5.5 Determinants……………………………………………....................................(31-32)

6. Conic Section……………….……..........................................................................(33)

7. Probability……………….……...........................................................................(34-35) Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 2

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1. Trigonometry 1.1 Trigonometrical Ratios and Identities 1. sin 2   cos 2   1

2. sec 2   1  tan 2 

3. cos ec 2  1  cot 2 

4. tan  

sin  cos

5. cot  

cos sin 

6. sin  

1 cos ec

7. cos 

1 sec

8. tan  

1 cot 

Addition and Subtraction Formulae For any two angles A and B 1. Sin A  B   sin A cos B  cos A sin B

2. Sin A  B   sin A cos B  cos A sin B

3. cos A  B   cos A cos B  sin A sin B

4. cos A  B   cos A cos B  sin A sin B

5. tan  A  B  

tan A  tan B 1  tan A. tan B

6. tan  A  B  

tan A  tan B 1  tan A. tan B

Double Angle Formulae 2. cos 2  cos 2   sin 2   1 2 sin 2   2 cos 2   1

1. sin 2  2 sin  cos , 3. tan 2 

2 tan  1  tan 2 

Triple angle Formulae 1. sin 3  3 sin   4 sin 3  3. tan 3 

2. cos 3  4 cos3   3 cos

3 tan   tan 3  1  3 tan 2 

Trigonometric Ratios of θ/2 1. sin   2 sin

  cos , 2 2

2. cos  cos 2

     sin 2  2 cos 2  1  2 sin 2 2 2 2 2

 2 3. tan   2  1  tan 2 2 tan

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Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 3

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Formulae for sin2θ & cos2θ in terms of tanθ 1. sin 2 

2 tan  1  tan 2 

2. cos 2 

1  tan 2  1  tan 2 

Formulae for sinθ & cosθ in terms of tanθ/2

 2 1. sin    1  tan 2 2

 2 2. cos   1  tan 2 2 1  tan 2

2 tan

Transformation of sum/differences into Products

C  D  C  D  1. sin C  sin D  2 sin   cos   2   2  C  D C  D 2. sin C  sin D  2 cos  sin    2   2  C  D C  D 3. cos C  cos D  2 cos  cos   2   2  C  D C  D C  D  DC  4. cos C  cos D  2 sin  sin    2 sin   sin   2   2   2   2  Transformations of Products into sum/difference 1. 2 SinA cos B  Sin( A  B)  Sin( A  B) 2. 2 cos A sin B  Sin( A  B)  Sin( A  B) 3. 2 cos A cos B  cos( A  B)  cos( A  B) 4. 2 sin A sin B  cos( A  B)  cos( A  B)

Trigonometric Ratios of (-θ) 1. sin      sin 

2. cos    cos

3. tan      tan 

4. cot    

5. sec     cot 

6. cos ec     cos ec

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Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 4

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

  Trigonometric Ratio of     : (All Positive) 2     1. cos     sin  2 

  2. sin      cos 2 

  3. tan      cot  2 

  4. cot     tan  2 

  5. cos ec     sec 2 

  6. sec     cos ec 2 

  Trigonometric Ratio of     :( Only sin  and cos ec is Positive) 2    1. cos      sin  2 

  2. sin      cos 2 

  3. tan      cot  2 

  4. cot      tan  2 

  5. cos ec     sec 2 

  6. sec      cos ec 2 

Trigonometric Ratios of     :( Only sin  and cos ec is Positive) 1. cos      cos

2. sin      sin 

3. tan       tan 

4. cot      cot 

5. cos ec     cos ec

6. sec      sec

Trigonometric Ratios of     :( Only tan  and cot  is Positive) 1. cos      cos

2. sin       sin 

3. tan      tan 

4. cot     cot 

5. cos ec      cos ec

6. sec      sec

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Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 5

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

 3  Trigonometric Ratio of     :( Only tan  and cot  is Positive) 2    3       sin  1. cos  2 

 3       cos 2. sin  2 

 3  3. tan      cot   2 

 3  4. cot     tan   2 

 3       sec 5. cos ec  2 

 3       cos ec 6. sec  2 

 3  Trigonometric Ratio of     :( Only cos and sec is Positive)  2   3      sin  1. cos  2 

 3       cos 2. sin  2 

 3  3. tan      cot   2 

 3  4. cot      tan   2 

 3       sec 5. cos ec  2 

 3      cos ec 6. sec  2 

Trigonometric Ratios of 2    :( Only cos and sec is Positive) 1. cos2     cos

2. sin 2      sin 

3. tan 2      tan 

4. cot2      cot 

5. cos ec2      cos ec

6. sec2     sec

Trigonometric Ratios of 2    : (All Positive) 1. cos2     cos

2. sin 2     sin 

3. tan 2     tan 

4. cot2     cot 

5. cos ec2     cos ec

6. sec2     sec

Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 6

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Short-cut method to remember the Trigonometric ratios

 n  1. sin      sin   2   n       cos 2. cos 2  

when n is an even integer

 n       tan  3. tan   2 

 n       cos 4. sin  2 

 n  5. cos      sin   2 

when n is an odd integer

 n       cot  6. tan   2 

θ

0o 30o

sin θ

0

cos θ

tan θ

45 o

60o

1

3 2

1 2

1

2

1

3 2

0

1

2

1

0

1 2

1

90o 120o

3

∞

3 2

135o

1 2

1

1 2



 3

-1



2

3 cot θ

∞

3

1

1

0

1

3 sec θ

1

2

2

180o 270o 360 o

1 2

0

-1

0

3 2

-1

0

1

1

0

∞

0

 

-1

 3

∞

0

∞

-2

 2

2

-1

∞

1

∞

-1

∞

3 cosec θ ∞

2

3

3 ∞

2

150o

3 2

2 3

1

2

2

2

3

Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1.2 Inverse Circular Functions 1. sin 1 sin x   x

2. cos 1 cos x   x

3. tan 1 tan x   x

4. cot 1 cot x   x

5. sec 1 sec x   x

6. cos ec 1 cos ec   x

  9. secsec x   x

8. cos cos 1 x  x 10. cos ec cos ec 1 x  x

1 11. sin 1    cos ec 1 x  x

1 12. cos 1    sec 1 x  x

1 13. tan 1    cot 1 x  x

1 14. cot 1    tan 1 x x

1 15. sec 1    cos 1 x  x

1 16. cos ec 1    sin 1 x  x

17. sin 1  x    sin 1 x

18. cos 1  x     cos 1 x



7. sin sin 1 x  x





1

19. tan 1  x    tan 1 x 20. tan 1 x  cot 1 x 



19. sin 1 x  cos 1 x 

 2

 2

21. sec 1 x  cos ec 1 x 

 2

22. sin 1 1  x 2  cos 1 x

23. cos 1 1  x 2  sin 1 x

24. tan 1 x 2  1  sec 1 x

25. cot 1 x 2  1  cos ec 1 x

26. sec 1 1  x 2  tan 1 x

27. cos ec 1 1  x 2  cot 1 x





28. sin 1 2 x 1  x 2  2 sin 1 x





29. sin 1 3 x  4 x 3  3 sin 1 x

30. cos 1 4 x 3  3x  3 cos 1 x

 2x  31. tan 1   2 tan 1 x 2  1 x 

 3x  x 3    3 tan 1 x 32. tan 1  2   1  3x 

 x y   tan 1 x  tan 1 y 33. tan 1   1  xy 





 x y    tan 1 x  tan 1 y 34. tan 1   1  xy 

Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 8

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Some Important Expansions: 1. sin x  x 

x3 x5   ......... 3! 5!

2. sinh x  x 

x 3 x5   ......... 3! 5!

3. cos x  1 

4. cosh x  1 

x2 x4   ......... 2! 4!

1 2 5. tan x  x  x 3  x 5  ......... 3 15

x 2 x3 6. e  1  x    ......... 2! 3! x

7. e

x

x2 x4   ......... 2! 4!

x 2 x3  1 x    ......... 2! 3!

Some useful substitutions:Expressions

Substitution

Formula

Result

3x  4 x 3

x  sin 

3 sin   4 sin 3 

Sin3θ

4 x 3  3x

x  cos 

4 cos 3   3 cos

cos3θ

3x  x 3 1  3x 2

x = tan θ

3 tan   tan 3  1  3 tan 2 

tan3θ

2x 1 x2

x = tan θ

1 x2 1 x2

x = tan θ

1  tan 2  1  tan 2 

cos2θ

2x 1 x2

x = tan θ

2 tan  1  tan 2 

tan2θ

1  2 x2

x = sin θ

1  2 sin 2 

cos2θ

2x2 1

x = cos θ

2 cos 2   1

cos2θ

1  x2

x = sin θ

1  sin 2 

cos2θ

1 x2

x = cos θ

1  cos 2 

sin2θ

x2 1

x = sec θ

sec 2   1

tan2θ

x2 1

x = cosec θ

cos ec 2  1

cot2θ

1 x2

x = tan θ

1  tan 2 

sec2θ

1 x2

x = cot θ

1  cot 2 

cosec2θ

2 tan  1  tan 2 

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sin2θ

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 9

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2. Differential and integral Calculus 2.1 Differentiation

f  x  h   f x  h

1. f x   lim h0

2. f ' a   lim h0

f a  h   f a  h

3.

dy y  lim dx x 0 x

4.

d k   0 ; k is constant function dx

5.

d x   1 dx

6.

d dx

7.

d  1  dx  x n

9.

d  1  1   dx  x  x 2

 n   n 1  x

 x  2 1x

8.

d n x  nx n 1 ; n  N dx

10.

d sin x   cos x dx

 

11.

d cos x    sin x dx

12.

d tan x   sec 2 x dx

13.

d sec x   sec x. tan x dx

14.

d cos ecx    cos ecx. cot x dx

15.

d x a  a x log a; a  0, a  1 dx

16.

d x e  ex dx

17.

d log x   1 dx x

18.

d 1 sin 1 x  ; 1  x  1 dy 1 x2

19.

d 1 cos 1 x  ;1  x  1 dx 1 x2

20.

d 1 tan 1 x  ; xR dx 1 x2

21.

d 1 cot 1 x  ; xR dx 1 x2

22.

d 1 sec 1  : dx x x2 1

23.

d 1 cos ec 1 x  ; dx x x2 1

 



 







 













x 1

x 1

Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 10

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Rules of Differentiations 1. Addition Rule: If y = (u + v) then 

dy du dv   dx dx dx

2. Substations Rule: If y = (u - v) then  3. Product Rule: If y = uv then 

dy dv du u v dx dx dx

u dy 4. Quotient Rule: If y  then   v dx 5. If y = f(u) is u = g(x) then 6. If u = f(y) , then

dy du dv   dx dx dx

v

du dv u dx dx 2 v

dy dy du  . dx du dx

du du dy dy  .  f  y  dx dy dx dx

dy dx dy 1 dx .  1 or  where 0 dx dx dy dx dy dy

7.

Derivatives of composite functions 1.

d  f  x n  n f  x n1 . d  f  x  dx dx

2.

d dx

3.

d  1  1 d  .  f  x    2 dx  f  x    f  x  dx

4.

d sin f  x   cos f x . d  f  x  dx dx

5.

d cos f x    sin f  x . d  f  x  dx dx

6.

d tan f x   sec 2 f x . d  f  x  dx dx

7.

d cot f  x    cos ec 2 f x . d  f  x  dx dx

8.

d sec f x   sec f x  tan f  x . d  f  x  dx dx

9.

d cos ecf x    cos ecf x  cot f x . d  f x  dx dx

10.

d log f  x   1 . d  f  x  dx f  x  dx

11.





f x 

1

d  f x  2 f  x  dx

d f x  d a  a f  x  log a.  f  x  dx dx



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

Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 11

fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 12.

d f x d e  e f  x   f x  dx dx





13.

d  f  g x n  n f g  x n1  f  g x  d g  x  dx dx

Derivatives of composite functions 1.

d 1 d sin 1 f  x   .  f  x  2 dx 1   f  x  dx

2.

d 1 d cos 1 f x   .  f  x  2 dx 1   f x  dx

3.

d 1 d tan 1 f x   .  f  x  2 dx 1   f x  dx

4.

d 1 d cot 1 f  x   .  f x  2 dx 1   f  x  dx

5.

d sec 1 f  x   dx f x

6.

d 1 d cos ec 1 f  x   .  f  x  2 dx f x   f x   1 dx

 















1





 f  x 

2

d  f x  dx 1 .



Implicit functions:Take the derivatives of these functions directly and find dy/dx Parametric functions:-

dy dy / dx  dx dx / dt

If x  f t  & y  g t  then

where

dx 0 dt g x

Logarithemic Differentiation:- If the function is in the form of  f x  Then taking Logarithm on both sides 15 & then find dy/dx Higher order Derivatives of composite functions:-

y2 

d2y  f " x  dx 2

In General; y n 

IInd order,

dny  f dx n

n

x

y3 

d3y  f " ' x dx 3

IIIrd order

nth order

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2.2 Limits Limits of function If for every   0 there exist δ > 0 such that if f  x   l   whenever 0  x  a   then we say lim of f(x) as x→ a is l i.e. lim f  x   l x a

Theorem of limits If f(x) and g(x) are two functions then 1. lim f  x   g x   lim f  x   lim g  x  x a

x a

x a

2. lim f  x   g x   lim f  x   lim g x  x a

x a

x a







3. lim f  x .g  x   lim f  x  . lim g x  x a

4. lim x a

x a

x a

f x  f  x  lim  x a g  x  lim g x  x a





5. limkf  x   k lim f  x  where k is constant x a

6. lim x a

x a

f  x   lim f x  x a

7. lim f  x 

p/q

x a





 lim f x  x a

p/q

: where p & q are integers

Some Important standard limits 1. lim x  a

2. lim c  c : where c is constant c  R

x a

3. lim x n  a n ; x a

x a

4. lim

nR

x a

sin  1  0 

6. lim

tan  1  0 

8. lim

 1   0 tan 

7. lim

 0

n  N,a  0

 1   0 sin 

5. lim

9. lim

xn  an  na n 1 ; xa

sin k k 

10. lim  0

tan k k 

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

x2 2 x 0 1  cos x

1  cos x 1  x 0 2 x

11. lim

12. lim

sin x 0  sin mx 0 m  ; lim  x 0 x 18 x 0 sin nx 0 n

1  cos kx k 2  x 0 2 x2

13. lim

15. lim cos x  1; x 0

14. lim

a x 1  log a where a > 0 x a x

ex 1 1 x a x

20. lim1  x 

1/ x

x 0

18. lim

log a 1  x   log a e x a x

log1  x   1; x a x

a 0

lim

lim1  kx 

1/ x

 e;

x  / 2

x 0

17. lim

19. lim

lim sin  / 2  1

16. lim sin x  0;

lim cos x  0

x  / 2

x 0

 ek

log1  kx  21. lim k x 0 x

cos ax  cos bx a 2  b 2 22. lim  2 x 0 cos cx  cos dx c d2

2 2  cos ax  cos bx  b  a 23. lim  x 0 2 x  

 1  ax  24. lim  x 0 1  bx  

 a  bx  25. lim  x 0 a  cx  

1/ x

e

bc a

26. lim

x

1 1  0; lim 2  0 2 x  x x   x

28. lim

x 

30. lim sin x  sin a

31. lim cos x  cos a

x a

x 1

1 1  0; lim  0 x   x x

lim k  k where k is constant

x 

32. lim

 e ab

1  0 where k >0 x  x k

27. lim

29. lim k  k ;

1/ x

x a

x  x 2  x 3  ...x n  n nn  1  x 1 2

ax  bx a  log ; 33. lim x 0 x b

a, b  0

sec x  1 1 cos ecx  1  ; lim 1 2 x 0 2 x0 x x2

34. lim

x

n

1    1 35. lim1  2   e x ; lim1    e x  n   x    n

h

 a 36. lim1    e a h  h

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2.3 Tangents and Normal Tangent at  x, y  to y  f x  Let y  f x  be a given curve and Px, y  and

x  x, y  y 

Q x  x, y  y  be two neighbouring points on it.

Q

Equation of the line PQ is

Yy

y  y  y  X  x  or Y  y  y  X  x  x  x  x x

P x, y 

This line will be tangent to the given curve at P if Q  P which in tern means that

x  0 and we know that limx0

y dy  x dx

Therefore the equation of the tangent is

Yy

dy X  x dx

Normal at x, y  The normal at x, y  being perpendicular to tangent will have its slope as  1 and dy dx hence its equation is

Yy

1 X  x dy dx

Geometrical meaning of dy dx

dy dx represents the slope of the tangent to the given curve y  f x  at any point x, y 



dy  tan dx

where  is the angle which the tangent to the curve makes with +ve direction of x-axis. dy  dy  In case we are to find the tangent at any point  x1 , y1  then   i.e. the value of dx  dx   x1 , y1 

at  x1 , y1  will represent the slope of the tangent and hence its equation in this case will be Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES  dy  y  y1    x  x1   dx   x1 , y1 

y  y1 

Normal

1 x  x1  dy dx  x1 , y1 

Condition for tangent to be parallel or perpendicular to x-axis If tangent is parallel to x-axis or normal is perpendicular to x-axis then

dy 0 dx If tangent is perpendicular to x-axis or normal is parallel to x-axis then

dx dy  0.   or dx dy 2.4 Maxima and Minima For the function y  f x  at the maximum as well as minimum point the tangent is parallel to x-axis so that its slope is zero. Calculate

dy dy  0 and solve for x. Suppose one root of  0 is at x=a. dx dx

If

d2y  ve for x=a, then maximum at x=a. d 2x

If

d2y   ve for x=a, then minimum at x=a. d 2x

If

d2y d3y  0 at x=a, then find . d 2x d 3x

If

d3y  0 at x=a, neither maximum nor minimum at x=a. d 3x

d3y d4y If 3  0 at x=a, then find 4 . d x d x If

d4y d4y  0 i.e +ve at x=a, then y is minimum at x=a and if  0 i.e -ve at x=a, then y d 4x d 4x

is maximum at x=a and so on. Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 2.5 Integration Indefinite Integration If

d F x   c  f x  , then we say that F x   c is an indefinite integral or dx

antiderivative of f  x  and we write

 f xdx  F x  c Some standard Integrals

x n 1 1.  x dx  c n 1 n

3.

1



x

 n  1

dx  2 x  c

1

1  c n  1 n  1x n1

2.

x

4.

 x dx  log x  c

n

dx 

1

ax c log a

5.  e x dx  e x  c

6.  a x dx 

7.  cos x dx  sin x  c

8.  sin x dx   cos x  c

9.  sec 2 xdx  tan x  c

10.  cos ec 2 x dx   cot x  c

11.  sec x tan x dx  sec x  c

12.  cos ecx cot x dx   cos ecx  c

dx

dx

 tan 1 x  c

 sin 1 x  c

14.

 sec 1 x  c

16.  cosh x dx  sinh x  c

17.  sinh x dx  cosh x  c

18.  sec h 2 xdx  tanh x  c

13.



15. 

1 x

2

dx 2

x x 1

19.  cos ech 2 x dx   coth x  c

1 x

2

20.  sec hx tanh x dx   sec hx  c

21.  cos echx coth x dx   cos echx  c

22.  tan xdx  logsec x   c

23.  cot x dx  logsin x   c

24.  sec x dx  logsec x  tan x   c

x  25.  cos ecx dx  log  tan   c 2 

26.

a

2

dx 1 x  tan 1  c 2 a a x

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 27.

x

28. 

2

dx 1 x  a  log  c; 2 2a a  x  a 

if x  a

dx 1 a  x  log  c ; if x  a 2 2a a x  a  x  2



 x  log x  x 2  a 2  c or sinh 1    c a x2  a2  dx

30.  31. 



dx

29. 

2

x a

2

dx 2

a x

2





 x or cosh 1    c a

 log x  x 2  a 2  c  sin 1

x c a











x 2  a 2 dx 

x 2 a2 x  a2  log x  x 2  a 2 2 2



x 2  a 2 dx 

x 2 a2 x  a2  log x  x 2  a 2 2 2

34.



a 2  x 2 dx 

x a2 x a2  x2  sin 1  c 2 2 a

35.

x

32. 33.

dx x2  a2

a

36.

 sin 0



1 x 1 x sec 1  c   cos ec 1  c a a a a

mx nx a sin dx   mn L L 2

 0,

mn

a  , 2

mn

Rules of Integration 1.   f1  x   f 2  x dx   f1 x dx   f 2 x dx 2.  k  f  x dx k  f  x dx , where k is constant 3.

 k f x  k f xdx  k  f x dx  k  f xdx , where k1 and k2 are constants 1 1

2

2

1

1

2

2

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Rule of integration by substitution 1. If x   t  ,

dx

 f x dx   f x  dt dt   f  t  ' t dt g ax  b  c a

2.

 f ax  b dx 

3.

 f x    f x  f ' x dx 

4.

 f x dx  log f x  c

n 1

n

 c : n  1

n 1

f ' x

Rules of integration by partial fraction This method can be used to evaluate an integral of the type

Px 

 Qx  dx

where (i) P(x) & Q(x) are Polynomials in x (ii) Degree of P(x) < degree of Q(x) (iii) Q(x) contains two/more distinct linear/quadratic factors i.e.

P x  A B C    Q x  a1 x  b1  a 2 x  b2  a3 x  b3   du



 uvdx  u  vdx    dx  vdx dx 2.5.1 Gamma integral 

(i) Gamma integral is given by

(n)   x n 1e x dx = (n  1) . 0

1     2 

(ii)

n  Bx 2

x e 0

dx 

1 2B

n 1 2

 n 1     2 

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 3. Differential Equations Order and degree of a differential equation The order of a differential equation is the order of the highest differential co-efficient present in the equation.

d2y dy Example: 2  2  3 y  0 is a second order differential equation. dx dx The degree of a differential equation is the degree of the highest derivative after removing the radical sign and fraction. 2

3

d2y  dy  Example:  2   2   3 y  0 has degree of 3.  dx   dx  D.E. of the first order and first degree 1.

Separation of the variables:

f  y dy    x dx 2.

Homogeneous Equation

dy f  x, y     x, y  is of the same degree. dx   x, y  if each term of f(x,y) and 3.

Equations reducible to homogeneous form x  X  h dy ax  by  c dy dY aX  bY  ah  bk  , let    y Y k dx Ax  By  C dx dX AX  BY  Ah  Bk

dy aX  bY ah  bk  c  0   Choose h, k so that Ah  Bk  C  0   dx AX  BY a b 1 Case of failure: A  B  m 4.



dy ax  by  c  dx max  by   C

Linear Differential Equations

dy  Py  Q where P and Q are function of x (but not y) or constant. dx

I .F .  e 

Pdx



y  I .F .   Q  I .F .dx  c

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Equations reducible to the linear form

dy  Py  Qy n dx 

1 divide by yn and put y n 1  z

1 dy 1  n 1 P  Q n y dx y



1  n  dy  dz yn

dx

dx

1 dz  Pz  Q 1  n dx



6. Exact differential Equation

M N Mdx + Ndy = 0 if y  x

 Mdx   terms of N not containing x  dy  C y constant

7.

Equations reducible to the exact form M N  x is a function of x alone, say f(x) then .F.  e  f  x  dx multiply with a) If y N

different equation.

M N  x is a function of y alone, say f(y) then .F.  e  f  y  dy . b) If y N

1 c) If M = yf1 (xy) and N = xf2 (xy), then .F .  Mx  Ny

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Linear D.E. of second order with constant coefficients

d2y dy  P  Qy  R where P, Q and R are function of x or constant. 2 dx dx y = C.F. + P.I. C.F. a) roots, real and different

y  C1 e m1x  C 2 e m2 x

b) roots, real and equal

y  C1  C 2 x e m2 x

c) roots imaginary

y  C1e    x  C 2 e   x  ex A cos  x  B sin x 

P.I.

1 1 ax ax a) f D  e  f a  e

if f a   0

then

1 1 e ax  x   e ax f D  f a 

1 1 n n b) f D  x   f D  x 1 1 1 1 c) f D 2 sin ax  f  a 2 sin ax and f D 2 cos ax  f  a 2 cos ax

 





 





1 1 If f(-a2) = 0 then f D 2 sin ax  f  a 2 sin ax

 





1 1 ax ax d) f D 2 e  x   e f D  a   x 

 

e)

1  x   e ax  e ax x  dx Da

1 1 n ax n f) f D  x sin ax  Im e f D  a  x

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 4. Vectors Cartesian coordinate system  Infinitesimal displacement dl  dxxˆ  dy yˆ  dz zˆ Volume element d  dxdydz  f f f Gradient: f  xˆ  yˆ  zˆ x y z Divergence:

  A A A . A  x  y  z x y z

Curl:

   A A   A A   A   z  y  xˆ   x  z z  x  z  y

Laplacian:

2 f 

 Ay Ax    yˆ   x  y  zˆ   

2 f 2 f 2 f   x 2 y 2 z 2

Spherical Polar Coordinate System( r , ,  )

x  r sin  cos  , y  r sin  sin  , z  r cos  z y r  x 2  y 2  z 2 ,   cos 1 ,   tan 1 r x  Infinitesimal displacement dl  drrˆ  rdˆ  r sin  dˆ 2 Volume element d  r sin  drd d

r range from 0 to  ,  from 0 to  , and  from 0 to 2 .

Gradient:

 f 1 f ˆ 1 f ˆ  f  rˆ    r r  r sin  

Divergence:

  1  1  1 A . A  2  r 2 Ar    sin  A   r r r sin   r sin  

Curl:

 rˆ    1    A  2 r sin   r   Ar

rˆ   rA

r sin ˆ        r sin  A 

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

Laplacian:

2 f 

1   2 f  1   f  1 r  2 sin   2 2     2 r r  r  r sin      r sin 

 2 f   2   

Cylindrical Coordinate System ( r ,  , z ) r  x2  y 2 ,

x  r cos  , y  r sin  , z  z and

  tan 1

y x

 Infinitesimal displacement d l  drrˆ  rdˆ  dz zˆ

Volume element

d  rdrd dz

r range from 0 to  ,  from 0 to 2 , and z from  to  . Gradient:

 f 1 f ˆ f f  rˆ    zˆ r r  z

Divergence:

  1  1 A A . A   rAr     z r r r  z

Curl:

 rˆ   1    A  r  r   Ar

Laplacian:

2 f 

rˆ   rA

zˆ     z   Az 

1   f  1  2 f  2 f  r   r r  r  r 2  2 z 2

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES VECTOR IDENTITIES Triple Product          (1) A.( B  C )  B.(C  A)  C.( A  B)          (2) A  ( B  C )  B( A.C )  C ( A.B) Product Rules    (3) ( fg )  f (g )  g ( f )                (4)  A.B  A  (  B)  B  (  A)  A. B  ( B.) A

 

 

      (5) .( f A)  f (. A)  A.( f )          (6) .( A  B)  B.(  A)  A.(  B)       (7)   ( f A)  f (  A)  A  ( f )                (8)   A  B  ( B. ) A  ( A.) B  A(.B)  B(. A)





Second Derivative    (9) .(  A)  0 i.e. divergence of a curl is always zero.   (10)   ( f )  0 i.e. curl of a gradient is always zero.        (11)   (  A)   . A   2 A

 

FUNDAMENTAL THEOREMS b

Gradient Theorem:





   f .dl  f  b   f  a  a

Divergence Theorem: Curl Theorem:

 

 

  . Ad   A.d a          A.d a   A.dl

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5. Algebra 5.1 Theory of Quadratic equations 1. Roots of the equation

 b  b 2  4ac 2a

ax 2  bx  c  0 are x 

Sum and Product of the roots If  and  be the roots, then

 

b c and  .  a a

2. To find the equation whose roots are  and  . The required equation will be

 x    x     0

or x 2     x   .  0 or x 2  Sx  P  0

where S is the sum and P is the product of the root. 3. Nature of the roots. Roots of the equation ax 2  bx  c  0 are x 

 b  b 2  4ac . 2a

The expression b 2  4 ac is called discriminant. (a) If b 2  4 ac  0 , roots are real. (i) If b 2  4 ac  0 , then roots are real and unequal.

 b  (ii) If b 2  4 ac  0 , then roots are real and equal   .  2a  (b) If b 2  4 ac  0 , then

b 2  4ac is imaginary. Therefore roots are imaginary and

unequal.

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5.2 Logarithms Properties of Logarithms ( a  0, a  1, m  0, n  0 ) 1. a x  y then x  log a y

2. log a a  1

3. log a 1  0

4. log b a 

5. Base changing formula log b a  log c a. log b c  6. log a mn  log a m  log a n ,

1 or log b a. log a b  1 log a b

log c a log c b

 n   log

log a m

a

m  log a n

7. log a m n  n log a m Or in particular log a a n  n  p p 8. log a q n q    log a n Or in particular log n q n q  q q

9. a log a n  n Rules of indices 1. a m  a n  a m n

 

3. a m

n

2.

am  a mn n a m

4. a  b   a m  b m

 a mn

m

am a 5.    m b b

6. a  m 

1 am

7. a 0  1

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5.3 Permutations and Combinations Permutation Each of the different arrangements which can be made by taking some or all of a number of things is called permutation. Combination Each of the different groups or selections which can be made by taking some or all of a number of things (irrespective of the order) is called combination. Fundamental Theorem If there are m ways of doing a thing and for each of the m ways there are associated n ways of doing a second thing then the total number of ways of doing the two things will be mn. Important Results (a) Number of permutations of n dissimilar things taken r at a time. n

Pr 

n!  nn  1n  2.......n  r  1 n  r !

where n! 1.2.3...............n . Note that n! n.n  1! n.n  1 . n  2 ! (b) Number of permutations of n dissimilar things taken all at a time. n

Pn 

n!  nn  1n  2 .......n  n  1 n  r !  nn  1n  2..........3.2.1  n!

(c) Number of combinations of n dissimilar things taken r at a time. n

Cr 

n P n!  r n  r !r! r!

(d) Number of combinations of n dissimilar things taken all at a time. n

Cn 

n! 1  1 n  n !n! 0!

 0!  1

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(e) If out of n things p are exactly alike of one kind, q exactly alike of second kind and r exactly alike of third kind and the rest all different, then the number of permutations of n things taken all at a time



n! p!.q!.r!

(f) If some or all of n things be taken at a time then the number of combinations will be

2n 1



n

C1  n C 2  .......... n C n  2 n  1

(g) n C r  n C n r

(h) n C r1  n C r2  r1  r2 or r1  r2  n .

(i) n C r  n C r 1  n 1C r

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5.4 Binomial Theorem (a) Statement of binomial theorem for positive and negative integral index

 x  a n

 x n  n C1 x n 1 a 1  n C 2 x n 2 a 2  ........ n C r x n r a r  ...... n C n 1 xa n 1  n C n a n

 x  a n

 x n  n C1 x n 1 a 1  n C 2 x n 2 a 2  ........ n C r x n r  a r  .................





(b) Number of terms and middle term n

The number of terms in the expansion of x  a  is n  1 .

n  If n is even there will be only one middle term i.e.   1th . 2 

 n  1  n 3 If n is odd there will be two middle terms i.e.  th and  th .  2   2  Expansion 2

2. a  b   a 2  2ab  b 2

2

3

4. a  b   a 3  3aba  b   b 2

1. a  b   a 2  2ab  b 2

3

3. a  b   a 3  3aba  b   b 3 Factorization



1. a 2  b 2  a  b a  b 

  a  b a  a  b a

3. a 3  b 3  a  b  a 2  ab  b 2 5. a n  b n 6. a n  b n

n 1

n 1

2. a 3  b 3  a  b  a  ab  b 2







4. a 4  b 4  a  b a  b  a  b 2



  ....

 a n 2 b  a n 3 b 2  ....  a n  2 b  a n 3 b 2

Sterling’s formula Using summation notation, binomial expansion can be written as

 x  y Sterling’s

n

n  n    x k y n k k 0  k 

approximation (or Sterling’s

formula)

is

an

approximation

for

large factorials. ln  n   n ln n  n where n is very large

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 5.5 Determinants In linear algebra the determinant is a value associated with a square matrix. The determinant of a matrix A is denoted by det( A) , or A . For instance, the determinant of the matrix a b a b  If A   then  ad  bc  c d c d

a b  If A   d e g h 

c a b  f  then det( A) = d e i  g h

c e f a h i

f i

b

d

f

g

i

c

d

e

g

h

Properties (a) The values of determinant is not altered by changing rows into columns and columns into rows.

e.g.

1 x

1 y

1 1 z =1

x y

x2 z

x2

y2

z2

z

z2

1

(b) If any two adjacent rows or two adjacent columns of a determinant are interchanged the determinant retains its absolute value but changes its sign.

e.g.

1 x

1 y

1 x z = 1

y 1

z 1

x2

y2

z2

y2

z2

x2

(c) If any two rows or two columns of determinant are identical then the determinant vanishes. Thus a1

c1

c1

a2

c2

c2

a3

c3

c3

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES

(d) If each constituent in any row or in any column be multiplied by the same factor then the determinant is multiplied by that factor pa1

b1

c1

a1

b1

c1

pa 2

b2

c 2  p a2

b2

c2 and qa 2

pa3

b3

c3

b3

c3

a3

a1 ra 3

b1

c1

a1

b1

c1

qb2

q c2  qr a 2

b2

c2

rb3

rc3

b3

c3

a3

(e) If each constituent in any row or in any column consists of r terms then the determinant can be expressed as the sum of r determinants. a1  1

b1

c1

a1

b1

Thus a 2  2

b2

c2  a2

b2

a 3  3

b3

c3

b3

a3

1 c2   2 c3 3

c1

b1

c1

b2

c2

b3

c3

(f) If one row or column is k times the other row or columns respectively then determinant of matrix will be 0 .

e.g.

a d

k .a k .d

c a b c f  0 and k .a k.b k.c  0

g

k .g

i

g

h

i

Some basic properties of determinants are: 1. det( I n )  I where I n is the n  n identity matrix. 2. det( AT )  det( A) where AT is transpose of A . 3. det( A1 ) 

1 where A1 is inverse of A . det( A)

4. For square matrices A and B of equal size, det( AB)  det( A) det( B) 5. det(cA)  c n det( A) for an n  n matrix Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: [email protected]

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 6. Conic Section In the Cartesian coordinate system the graph of a quadratic equation of two variables represent a conic section which is given by Ax 2  Bxy  Cy 2  Dx  Ey  F  0 . The conic sections described by this equation can be classified with the discriminant

D  B 2  4 AC 

if D  0 , the equation represents an ellipse



if D  0 , A  C and B  0 , the equation represents a circle which is a special case of an ellipse;



if D  0 , the equation represents a parabola



if D  0 the equation represents a hyperbola



if we also have D  0 , A  C  0 , the equation represents a rectangular hyperbola

Note that A and B are polynomial coefficients, not the lengths of semi-major/minor axis as defined in some sources. Conic

Equation

Eccentricity

section

x2  y 2  a2

Ellipse

x2 y 2  1 a 2 b2

Hyperbola

Polar equation

Parametric form

rectum

Circle

Parabola

Semi-lactus

a

0

e  1

b2 a2

y 2  4ax

e 1

x2 y 2  1 a2 b2

b2 e  1 2 a

l

b2 a

2a

l

b2 a

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ra

x  a cos  , y  a sin 

l  1  e cos  r 0  e 1

x  a cos  , y  b sin 

l  1  cos  r

x  at 2 , y  2at

l  1  e cos  r e 1

x  a tan  , y  b sec 

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 7. Probability

Probability The probability pr of occurrence of an event r in a system is defined with respect to statistical ensemble of N such a systems. If N r systems in the ensemble exhibit the event r then

pr 

Nr N

Probability density The probability density  (u ) is defined by the property that  (u )du yields the probability of finding the continuous variable u in the range between u and u  du .

Mean value The mean value of u is denoted by u as defined as u    pr ur where the sum is over r

all possible value values u r of the variable u and pr is denotes the probability of occurrence of the particular value u r .Above definition is for discrete variable . For continuous variable u , u =  u  (u )du

Dispersions or variance The dispersion of u is defined as  2  ( u ) 2    pr (ur  u  )2 which is equivalent r

to  2   ( u ) 2    (u 2   u 2 ) r

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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Joint probability If both events A and B occur on a single performance of an experiment, this is called the intersection or joint probability of A and B, denoted as

p ( A  B) .

Independent probability If two events, A and B are independent then the joint probability is

p ( A  B)  p ( A).P( B) Mutually exclusive If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as p ( A  B ) . If two events are mutually exclusive then the probability of either occurring is

p ( A  B)  p ( A)  P ( B)

Not mutually exclusive If the events are not mutually exclusive then p ( A  B )  p ( A)  P ( B )  p ( A  B )

Conditional probability Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written p ( A / B ) , and is read "the probability of A, given B". It is defined by

p ( A / B) 

p ( A  B) p ( B)

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Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 35