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Engineering Mathematics Formulae Sheet 1. Change of Variable of Integration in 2D ZZ ZZ f (x(u, v ), y(u, v ))|J(u, v )| dudv f (x, y) dxdy = R∗

R

2. Transformation to Polar Coordinates x = r cos θ,

y = r sin θ,

J(r, θ ) = r

3. Change of Variable of Integration in 3D ZZZ ZZZ F (u, v, w)|J(u, v, w)| dudvdw f (x, y, z) dxdydz = V∗

V

4. Transformation to Cylindrical Coordinates x = r cos θ,

y = r sin θ,

z = z,

J(r, θ, z ) = r

5. Transformation to Spherical Coordinates x = r cos θ sin φ,

y = r sin θ sin φ,

z = r cos φ,

J(r, θ, φ) = r 2 sin φ

6. Line Integrals Z

f (x, y, z) ds = C

Z

b

f (x(t), y (t), z (t)) a

p

x′ (t)2 + y ′ (t)2 + z ′ (t)2 dt

7. Work Integrals Z

F(x, y, z) · dr = C

Z

a

b

F1

dz dy dx + F2 + F3 dt dt dt dt

8. Surface Integrals ZZ ZZ q g(x, y, f (x, y)) fx2 + fy2 + 1 dxdy g(x, y, z) dS = S

R

9. Flux Integrals

For a surface with upward unit normal, ZZ

F·n ˆ dS = S

ZZ

−F1 fx − F2 fy + F3 dydx R

10. Gauss’ (Divergence) Theorem ZZZ

∇ · F dV =

V

ZZ

F·n ˆ dS S

11. Stokes’ Theorem ZZ

(∇ × F) · n ˆ dS = S

Z

F · dr C

12. Complex Exponential Formulae sinh x = 12 (ex − e−x ) cosh x = 21 (ex + e−x ) eix = cos x + i sin x sin z = 2i1 (eiz − e−iz ) cos z = 12 (eiz + e−iz ) 13. Trigonometric and Hyperbolic Formulae cos2 x + sin2 x = 1 1 + tan2 x = sec2 x cot2 x + 1 = cosec2 x

cosh2 x − sinh2 x = 1 1 − tanh2 x = sech2 x coth2 x − 1 = cosech 2 x

cos 2x = cos2 x − sin2 x cos 2x = 2 cos2 x − 1 cos 2x = 1 − 2 sin2 x sin 2x = 2 sin x cos x

cosh 2x = cosh2 x + sinh2 x cosh 2x = 2 cosh2 x − 1 cosh 2x = 1 + 2 sinh2 x sinh 2x = 2 sinh x cosh x

sin(x + y) = sin x cos y + cos x sin y sin x cos y = 21 [sin(x − y) + sin(x + y)] sin x sin y = 21 [cos(x − y) − cos(x + y)]

cos(x + y) = cos x cos y − sin x sin y

14. Standard Integrals R sin x dx = − cos x + C

cos x cos y = 21 [cos(x − y) + cos(x + y )]

R

cos x dx = sin x + C

R

sec x dx = log | sec x + tan x| + C

R

cosec x dx = log | cosec x − cot x| + C

R

sec2 x dx = tan x + C

R

cosec2 x dx = − cot x + C

R

sinh x dx = cosh x + C

R

cosh x dx = sinh x + C

R

sech2 x dx = tanh x + C

R

cosech2 x dx = − coth x + C

R

√ 1 a2 −x2

dx = arcsin

R

1 a2 +x2

R

√ 1 x2 −a2

dx = arccosh

R

√ 1 x2 +a2

 x a

+C

x a

+C

arctan

 x

+C

dx = arcsinh

x

+C

dx =

1 a

a

a

where a > 0 is constant and C is an arbitrary constant of integration....