Formula sheet - Summary Advanced Engineering Mathematics PDF

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1. The alternative notation for the exponential function ex = exp(x). 2. Euler’s formula: ejx = cos x + j sin x 3. De Moivre’s formulae: z n  re j

1







1

1

j

  2 k

z n  re   2  n  r n e n .  r ne jn ; f f 4. Directional derivative: Dn f  m  cos   sin  , where  is the angle between the unit x y vector n and axis x. f f f 5. The gradient: grad f  x, y , z   f  i  j  k . x y z i  ay j  az k a a x 6. The unit vector n parallel to vector a: n   . 2 2 2 a ax  a y  az n

j

k

7. The integrating factor for a linear ODE: y  p  x  y  q  x ; I x   exp  p x dx . 8. The Taylor series: f  x   f  a   f   a  9. The Fourier series: f  t  

2 an  T

a0  2

 x  a x a  f   a   1! 2! 2



 a cos  n t   b sin  n t , n

n

a

 x  a n n!

.

where  = 2  /T and

n 1

T 2

2 T

 f  t  cos  n t  dt,  n  0,1, 2,

T 2

T 2

T 2

10. General solution to the linear homogeneous 2-nd order ODE with constant coefficients: px p x a) y  Ae 1  Be 2 – if roots are distinct and real; x b) y  e C1 cos x  C 2 sin x  – if roots are complex-conjugate (p1,2 =  ± i); rx c) y  C 1  C 2x e – if roots are equal (p1 = p2 = r). 11. The method of undetermined coefficients for a particular solution of non-homogeneous ODE: a) If function in the right-hand side is f (x) = ekxP(x), where P(x) is a polynomial of degree n, then the trial solution should be taken in the form ypn = ekxQ(x), where Q(x) is an n-th degree polynomial with the unknown coefficients. b) If function f (x) = ekxP(x) cos mx or f (x) = ekxP(x) sin mx, where P(x) is a polynomial of degree n, then the trial solution should be taken in the form ypn = ekxQ(x) cos mx + ekxR(x) sin mx, where Q(x) and R(x) are an n-th degree polynomials with the unknown coefficients. c) If any term of the particular trial solution ypn is a solution of a complementary homogeneous equation (the resonance case), then multiply ypn by x or even by x2 if necessary. 12. The chain rule for the function of 3 variables w = f (x, y, z), where x, y and z are in turn functions of other three variables s, t and u, x = x(s, t, u), y = y(s, t, u), z = z(s, t, u): w w x w y w z w w x w y w z w w x w y w z , , .          s x s y s z s t x t y t z t u x u y u z u 13. The criterion for the complete differential for a function of two variables (the criterion that the plane vector filed is potential): the expression df = P(x, y)dx + Q(x, y)dy is a complete differential if Py = Qx. Correspondingly the vector field V = P(x, y) i + Q(x, y) j is potential at the same condition. 14. Equation of a tangent plane to a surface z = f (x, y) at the point P(x0, y0, z0): z = z0 + f 'x (x0, y0)(x – x0) + f 'y (x0, y0)(y – y0). 15. The differential of function w = f (x, y, z) is: df = f 'x(x, y, z)dx + f 'y(x, y, z)dy + f 'z(x, y, z)dz. 16. Necessary conditions for the extremum of a differentiable function: all its partial derivatives must be zero at the point of extremum.

1

17. Sufficient conditions for the extremum. Find all second partial derivatives at the critical point: A = fxx(x0, y0), B = fxy(x0, y0), C = fyy(x0, y0) and calculate the discriminant: D = AC  B2. - If D > 0 and A > 0 – then function z = f (x, y) has a local minimum at the critical point (x0, y0); - If D > 0 and A < 0 – then function z = f (x, y) has a local maximum at the critical point (x0, y0); - If D < 0 – then function z = f (x, y) has a saddle at the critical point (x0, y0); - If D = 0 – then the test is inconclusive. 18. Area of the plane figure can be calculated by means of a double integral over the domain bounded by the given curves A = 1dA. 19. The links between the Cartesian and polar coordinates: y 2 2 x  r cos , y  r sin  ; r  x  y ,   atan . x 20. Conversion from the Cartesian to polar coordinates in double integrals:  f  x, y  dxdy   f r ,   rdrd . 

21. Green’s theorem:



 Q

P 

 F  x, y   dr   P  x, y  dx  Q  x, y  dy    x  y  dxdy. C



C

22. The area of a domain via the contour integral: A   F  x, y   dr  C

  curlF 

z

dxdy 



1dxdy. 

23. The line integral of the first kind is: T

 F  x, y, z  ds   F   t  ,  t  ,   t   

,

0

where Γ is a continuous line in 3D space: r (t) =  (t) i +  (t) j +  (t) k, where 0 ≤ t ≤ T, and scalar function F(x, y, z) is defined on  24. The line integral of the second kind is: A   F  x , y , z   dr    P  x , y , z  dx  Q  x , y , z dy  R x , y , z dz   



T

P  t  , t ,  t  



0

where Γ is a continuous line in 3D space: r (t) =  (t) i +  (t) j +  (t) k, where 0 ≤ t ≤ T, and F(x, y, z) is the vector field in 3D space: F(x, y, z) = P(x, y, z) i + Q(x, y, z) j + R(x, y, z) k. 25. The characteristic equation for eigenvalues: det (A ‒  I ) = 0; 26. Equation for eigenvectors of a matrix A: Av =  v, where  is a known eigenvalue. 27. Definition of the symmetric matrix AT = A. 28. The norm of n-component vector: v  12   22  . u11 u 2 2  u v 29. The angle between two vectors: cos    u v u12  u22 

.

30. Symmetric matrix: AT = A. Orthogonal matrix P: PT = P ‒1 or equivalently, if PPT = PTP = I. 31. sin 2 = 2 sin cos ; cos 2 = cos2 – sin2 ; 2sin2  = 1 – cos 2; 2cos2 = 1 + cos 2 . b

32. Integration by parts:

b

a

33.

a

2

b

 u (x ) (x )dx  u (x ) (x ) a   (x )u (x )dx .

dx x 1  tan  1 ; 2 a a x

a

a

2

dx a x 1 ;  ln 2 2a a  x x

2



dx a x 2

2

 sin  1

x x  cos  1 . a a...