Mathematical Physics

Question 1

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

Answer 1

e1 = 𝐢
e2 = 𝐣
e3 = 𝐤

Question 2

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

Answer 2

e1 = 𝐞𝜌
e2 = 𝐞𝜙
e3 = 𝐞𝑧

(poz)

Question 3

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

Answer 3

e1 = 𝐞𝑟
e2 = 𝐞𝜃
e3 = 𝐞𝜙

(rot - sphere, rotting apple)

Question 4

What is the name for h1, h2 and h3?

Answer 4

Line elements

or

Coordinate scale factors

Question 5

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

Answer 5

h1 = 1
h2 = 1
h3 = 1

Question 6

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

Answer 6

h1 = 1
h2 = 𝜌
h3 = 1

Question 7

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

Answer 7

h1 = 1
h2 = r
h3 = r sin𝜃

Question 8

What is the product rule?

Answer 8

udv + vdu

Question 9

What is the chain rule?

Answer 9

dy/dx = dy/du × du/dx

Question 10

What is the quotient rule?

Where f(x) = u/v

Answer 10

(vdu - udv) / v^2

Question 11

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

Answer 11

a . b = a1b1 + a2b2 + a3b3

Question 12

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

Answer 12

(a . b) / |b|

Question 13

What is the characteristic polynomial of a matrix?

Answer 13

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

det (λI − A)

Question 14

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

Answer 14

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

Find determinant

Question 15

What is the formula for integration by parts?

Answer 15

∫ u dv = u v - ∫ v du

Question 16

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

Answer 16

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

Question 17

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

Answer 17

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

Question 18

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

Answer 18

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

Question 19

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

Answer 19

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

Question 20

cos(2x) = ?

Answer 20

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

Question 21

sin(2x) = ?

Answer 21

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

Question 22

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

Answer 22

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

Question 23

What is the Real Fourier Sum (RFS)?

Answer 23

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

Question 24

What is a0?

Answer 24

a0 = 1/π ∫(π, - π) f(x) dx

Question 25

What is a(n)?

Answer 25

a(n) = 1/π ∫(π, - π) f(x) cos(nx) dx

Question 26

What is b(n)?

Answer 26

b(n) = 1/π ∫(π, - π) f(x) sin(nx) dx

Question 27

sin(π) = ?

Answer 27

sin(π) = 0

Question 28

cos(π) = ?

Answer 28

cos(π) = -1

Question 29

sin(0) = ?

Answer 29

sin(0) = 0

Question 30

cos(0) = ?

Answer 30

cos(0) = 1

Question 31

cos(-π) = ?

Answer 31

cos(-π) = -1

Question 32

sin(-π) = ?

Answer 32

sin(-π) = 0

Question 33

cos(2π) = ?

Answer 33

cos(2π) = 1

Question 34

Euler’s Formula

What is e^(inx) equal to?

Answer 34

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

Question 35

Euler’s Formula

What is e^(-inx) equal to?

Answer 35

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

Question 36

What is e^(-∞)?

Answer 36

e^(-∞) = 0

Question 37

What is e^(0)?

Answer 37

e^(0) = 1

Question 38

sin(2π) = ?

Answer 38

sin(2π) = 0

Question 39

cos(π/2) = ?

Answer 39

cos(π/2) = 0

Question 40

sin(π/2) = ?

Answer 40

sin(π/2) = 1

Question 41

What is the diffusion equation?

Answer 41

∇(^2) u = ( 1/α^2 ) ( ∂u / ∂t )

Question 42

What is the laplace equation?

Answer 42

∇^2 T = 0

Where T is a scalar function.

Question 43

Laplace Equation

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

Answer 43

When the plate is in the y-direction,

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

Question 44

Laplace Equation

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

Answer 44

When the plate is in the x-direction,

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

Question 45

What are the general solutions to the diffusion equation?

Answer 45

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

Question 46

Diffusion Equation

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

Answer 46

∂u/∂x|x=0 = 0

Question 47

Diffusion Equation

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

Answer 47

∂u/∂x |x=L = 0

Question 48

Diffusion Equation

Give the condition for k for an insulated bar.

Answer 48

Select cos(kx) solutions

with the condition that

kL = nπ

Question 49

Diffusion Equation

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

Answer 49

u(0, t) = 0

Question 50

Diffusion Equation

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

Answer 50

u(L, t) = 0

Question 51

Diffusion Equation

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

Answer 51

Select sin(kx) solutions

with the condition that

kL = nπ

Question 52

What is the wave equation?

Answer 52

∇^2 φ = ( 1 / c^2 ) ( ∂^2 φ / ∂ t^2 )

Question 53

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

Answer 53

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

Question 54

Wave Equation

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

Answer 54

y(0, t) = 0

y(L, t) = 0

Question 55

Wave Equation

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

Answer 55

∂y(0, t) / ∂x = 0

∂y(L, t) / ∂x = 0

Question 56

Wave Equation

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

Answer 56

∂y(0, t) / ∂x = 0

y(L, t) = 0

Question 57

Wave Equation

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

Answer 57

y(0 , t) = 0

∂y(L, t) / ∂x = 0

Question 58

Wave Equation

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

Answer 58

sin(kx)

Question 59

Wave Equation

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

Answer 59

cos(kx)

Question 60

Wave Equation

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

Answer 60

cos(kx)

Question 61

Wave Equation

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

Answer 61

sin(kx)

Question 62

Wave Equation

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

Answer 62

kL = nπ

Question 63

Wave Equation

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

Answer 63

kL = nπ

Question 64

Wave Equation

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

Answer 64

kL = (n + 1/2) π

Question 65

Wave Equation

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

Answer 65

kL = (n + 1/2) π

Question 66

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

Answer 66

Upper: r = a

Lower: r = 0

Question 67

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

Answer 67

Upper: θ = π

Lower: θ = 0

Question 68

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

Answer 68

Upper: φ = 2π

Lower: φ = 0

Question 69

What is the general equation for the volume integral?

Answer 69

𝛿𝑉 = ℎ1 ℎ2 ℎ3 𝛿𝑥1 𝛿𝑥2 𝛿𝑥3

Question 70

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

Answer 70

𝛿𝑉 = 𝑟^2 sin 𝜃 𝛿𝑟 𝛿𝜃 𝛿φ

Question 71

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

Answer 71

𝛿𝑉 = 𝜌 𝛿𝜌 𝛿𝜙 𝛿z

Question 72

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

Answer 72

𝛿𝑉 = 𝛿𝑥 𝛿𝑦 𝛿z

Question 73

What is C(0)?

Answer 73

1/2π ∫(π, - π) f(x) dx

Question 74

What is C(n)?

Answer 74

1/2π ∫(π, - π) f(x) e^(-inx) dx