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Engineering Physics Year 2 > Mathematical Physics > Flashcards

Mathematical Physics Flashcards

1
Q

What are the values for e1, e2 and e3 for Cartesian Coordinates?

A

e1 = 𝐢
e2 = 𝐣
e3 = 𝐤

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2
Q

What are the values for e1, e2 and e3 for Cylindrical Polar Coordinates?

A

e1 = 𝐞𝜌
e2 = 𝐞𝜙
e3 = 𝐞𝑧

(poz)

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3
Q

What are the values for e1, e2 and e3 for Spherical Polar Coordinates?

A

e1 = 𝐞𝑟
e2 = 𝐞𝜃
e3 = 𝐞𝜙

(rot - sphere, rotting apple)

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4
Q

What is the name for h1, h2 and h3?

A

Line elements

or

Coordinate scale factors

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5
Q

What are the values for h1, h2 and h3 for Cartesian Coordinates?

A

h1 = 1
h2 = 1
h3 = 1

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6
Q

What are the values for h1, h2 and h3 for Cylindrical Polar Coordinates?

A

h1 = 1
h2 = 𝜌
h3 = 1

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7
Q

What are the values for h1, h2 and h3 for Spherical Polar Coordinates?

A

h1 = 1
h2 = r
h3 = r sin𝜃

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8
Q

What is the product rule?

A

udv + vdu

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9
Q

What is the chain rule?

A

dy/dx = dy/du × du/dx

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10
Q

What is the quotient rule?

Where f(x) = u/v

A

(vdu - udv) / v^2

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11
Q

Where a and b are vectors, what is a . b?

A

a . b = a1b1 + a2b2 + a3b3

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12
Q

Where a and b are vectors, what is the projection of a in the direction of b?

A

(a . b) / |b|

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13
Q

What is the characteristic polynomial of a matrix?

A

Where A is a matrix, the characteristic polynomial is given by:

det (λI − A)

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14
Q

Where a and b are vectors, what is a x b?

A

|i, j, k|
|ax, ay, az|
|bx, by, bz|

Find determinant

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15
Q

What is the formula for integration by parts?

A

∫ u dv = u v - ∫ v du

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16
Q

1/2π ∫(π, - π) Σ a(n) cos(nx) cos(mx) dx = ?

A

1/2π ∫(π, - π) Σ a(n) cos(nx) cos(mx) dx = a(m)/2

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17
Q

1/2π ∫(π, - π) Σ b(n) sin(nx) sin(mx) dx = ?

A

1/2π ∫(π, - π) Σ b(n) cos(nx) cos(mx) dx = b(m)/2

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18
Q

1/2π ∫(π, - π) Σ b(n) sin(nx) cos(mx) dx = ?

A

1/2π ∫(π, - π) Σ b(n) sin(nx) cos(mx) dx = 0

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19
Q

1/2π ∫(π, - π) Σ b(n) cos(nx) sin(mx) dx = ?

A

1/2π ∫(π, - π) Σ b(n) cos(nx) sin(mx) dx = 0

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20
Q

cos(2x) = ?

A

cos(2x) = cos^2(x) - sin^2(x)

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21
Q

sin(2x) = ?

A

sin(2x) = 2sin(x)cos(x)

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22
Q

cos^2(x) + sin^2(x) = ?

A

cos^2(x) + sin^2(x) = 1

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23
Q

What is the Real Fourier Sum (RFS)?

A

f(x) = a0/2 + Σ a(n) cos(nx) + Σ b(n) sin(nx)

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24
Q

What is a0?

A

a0 = 1/π ∫(π, - π) f(x) dx

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25
Q

What is a(n)?

A

a(n) = 1/π ∫(π, - π) f(x) cos(nx) dx

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26
Q

What is b(n)?

A

b(n) = 1/π ∫(π, - π) f(x) sin(nx) dx

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27
Q

sin(π) = ?

A

sin(π) = 0

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28
Q

cos(π) = ?

A

cos(π) = -1

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29
Q

sin(0) = ?

A

sin(0) = 0

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30
Q

cos(0) = ?

A

cos(0) = 1

31
Q

cos(-π) = ?

A

cos(-π) = -1

32
Q

sin(-π) = ?

A

sin(-π) = 0

33
Q

cos(2π) = ?

A

cos(2π) = 1

34
Q

Euler’s Formula

What is e^(inx) equal to?

A

e^(inx) = cos(nx) + i sin(nx)

35
Q

Euler’s Formula

What is e^(-inx) equal to?

A

e^(-inx) = cos(nx) - i sin(nx)

36
Q

What is e^(-∞)?

A

e^(-∞) = 0

37
Q

What is e^(0)?

A

e^(0) = 1

38
Q

sin(2π) = ?

A

sin(2π) = 0

39
Q

cos(π/2) = ?

A

cos(π/2) = 0

40
Q

sin(π/2) = ?

A

sin(π/2) = 1

41
Q

What is the diffusion equation?

A

∇(^2) u = ( 1/α^2 ) ( ∂u / ∂t )

42
Q

What is the laplace equation?

A

∇^2 T = 0

Where T is a scalar function.

43
Q

Laplace Equation

What are the general solutions when the plate is in the y-direction?

A

When the plate is in the y-direction,

T(x, y) = { e^ky } { cos(kx) }
________{ e^-ky } { sin(kx) }

44
Q

Laplace Equation

What are the general solutions when the plate is in the x-direction?

A

When the plate is in the x-direction,

T(x, y) = { e^kx } { cos(ky) }
________{ e^-kx } { sin(ky) }

45
Q

What are the general solutions to the diffusion equation?

A

u(x, t) = { cos(kx) } { e^(-k^2) (α^2) t }
{ sin(kx) }

46
Q

Diffusion Equation

Give the boundary condition at x = 0 for an insulated bar.

A

∂u/∂x|x=0 = 0

47
Q

Diffusion Equation

Give the boundary condition for x = L for an insulated bar.

A

∂u/∂x |x=L = 0

48
Q

Diffusion Equation

Give the condition for k for an insulated bar.

A

Select cos(kx) solutions

with the condition that

kL = nπ

49
Q

Diffusion Equation

Give the boundary condition at x = 0 for a bar held at 0 degrees for t > 0.

A

u(0, t) = 0

50
Q

Diffusion Equation

Give the boundary condition at x = L for a bar held at 0 degrees for t > 0.

A

u(L, t) = 0

51
Q

Diffusion Equation

Give the condition for k for a bar held at 0 degrees at both edges.

A

Select sin(kx) solutions

with the condition that

kL = nπ

52
Q

What is the wave equation?

A

∇^2 φ = ( 1 / c^2 ) ( ∂^2 φ / ∂ t^2 )

53
Q

What are the general solutions to the wave equation for y(x, t)?

A

y(x, t) = { cos(kx) } {cos(kct) }
________{ sin(kx) } { sin(kct) }

54
Q

Wave Equation

What are the boundary conditions for a stretched string clamped at x = 0 and x = L?

A

y(0, t) = 0

y(L, t) = 0

55
Q

Wave Equation

What are the boundary conditions for a tube of length L open at both ends?

A

∂y(0, t) / ∂x = 0

∂y(L, t) / ∂x = 0

56
Q

Wave Equation

What are the boundary conditions for a tube of length L open at x = 0 and closed at x = L?

A

∂y(0, t) / ∂x = 0

y(L, t) = 0

57
Q

Wave Equation

What are the boundary conditions for a tube of length L closed at x = 0 and open at x = L?

A

y(0 , t) = 0

∂y(L, t) / ∂x = 0

58
Q

Wave Equation

Which general solutions are selected for a stretched string clamped at x = 0 and x = L?

A

sin(kx)

59
Q

Wave Equation

Which general solutions are selected for a tube of length L open at both ends?

A

cos(kx)

60
Q

Wave Equation

Which general solutions are selected for a tube of length L open at x = 0 and closed at x = L?

A

cos(kx)

61
Q

Wave Equation

Which general solutions are selected for a tube of length L closed at x = 0 and open at x = L?

A

sin(kx)

62
Q

Wave Equation

What is the condition for k when a stretched string is clamped at x = 0 and x = L?

A

kL = nπ

63
Q

Wave Equation

What is the condition for k when a tube of length L is open at both ends?

A

kL = nπ

64
Q

Wave Equation

What is the condition for k when a tube of length L is open at x = 0 and closed at x = L?

A

kL = (n + 1/2) π

65
Q

Wave Equation

What is the condition for k when a tube of length L is closed at x = 0 and open at x = L?

A

kL = (n + 1/2) π

66
Q

When finding the volume integral for spherical coordinates, what are the limits for r?

A

Upper: r = a

Lower: r = 0

67
Q

When finding the volume integral for spherical coordinates, what are the limits for θ?

A

Upper: θ = π

Lower: θ = 0

68
Q

When finding the volume integral for spherical coordinates, what are the limits for φ?

A

Upper: φ = 2π

Lower: φ = 0

69
Q

What is the general equation for the volume integral?

A

𝛿𝑉 = ℎ1 ℎ2 ℎ3 𝛿𝑥1 𝛿𝑥2 𝛿𝑥3

70
Q

What is the equation for the volume integral for spherical polar coordinates?

A

𝛿𝑉 = 𝑟^2 sin 𝜃 𝛿𝑟 𝛿𝜃 𝛿φ

71
Q

What is the equation for the volume integral for cylindrical polar coordinates?

A

𝛿𝑉 = 𝜌 𝛿𝜌 𝛿𝜙 𝛿z

72
Q

What is the equation for the volume integral for cartesian coordinates?

A

𝛿𝑉 = 𝛿𝑥 𝛿𝑦 𝛿z

73
Q

What is C(0)?

A

1/2π ∫(π, - π) f(x) dx

74
Q

What is C(n)?

A

1/2π ∫(π, - π) f(x) e^(-inx) dx

Engineering Physics Year 2 (6 decks)

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