Mathematical Physics Flashcards
What are the values for e1, e2 and e3 for Cartesian Coordinates?
e1 = π’
e2 = π£
e3 = π€
What are the values for e1, e2 and e3 for Cylindrical Polar Coordinates?
e1 = ππ
e2 = ππ
e3 = ππ§
(poz)
What are the values for e1, e2 and e3 for Spherical Polar Coordinates?
e1 = ππ
e2 = ππ
e3 = ππ
(rot - sphere, rotting apple)
What is the name for h1, h2 and h3?
Line elements
or
Coordinate scale factors
What are the values for h1, h2 and h3 for Cartesian Coordinates?
h1 = 1
h2 = 1
h3 = 1
What are the values for h1, h2 and h3 for Cylindrical Polar Coordinates?
h1 = 1
h2 = π
h3 = 1
What are the values for h1, h2 and h3 for Spherical Polar Coordinates?
h1 = 1
h2 = r
h3 = r sinπ
What is the product rule?
udv + vdu
What is the chain rule?
dy/dx = dy/du Γ du/dx
What is the quotient rule?
Where f(x) = u/v
(vdu - udv) / v^2
Where a and b are vectors, what is a . b?
a . b = a1b1 + a2b2 + a3b3
Where a and b are vectors, what is the projection of a in the direction of b?
(a . b) / |b|
What is the characteristic polynomial of a matrix?
Where A is a matrix, the characteristic polynomial is given by:
det (Ξ»I β A)
Where a and b are vectors, what is a x b?
|i, j, k|
|ax, ay, az|
|bx, by, bz|
Find determinant
What is the formula for integration by parts?
β« u dv = u v - β« v du
1/2Ο β«(Ο, - Ο) Ξ£ a(n) cos(nx) cos(mx) dx = ?
1/2Ο β«(Ο, - Ο) Ξ£ a(n) cos(nx) cos(mx) dx = a(m)/2
1/2Ο β«(Ο, - Ο) Ξ£ b(n) sin(nx) sin(mx) dx = ?
1/2Ο β«(Ο, - Ο) Ξ£ b(n) cos(nx) cos(mx) dx = b(m)/2
1/2Ο β«(Ο, - Ο) Ξ£ b(n) sin(nx) cos(mx) dx = ?
1/2Ο β«(Ο, - Ο) Ξ£ b(n) sin(nx) cos(mx) dx = 0
1/2Ο β«(Ο, - Ο) Ξ£ b(n) cos(nx) sin(mx) dx = ?
1/2Ο β«(Ο, - Ο) Ξ£ b(n) cos(nx) sin(mx) dx = 0
cos(2x) = ?
cos(2x) = cos^2(x) - sin^2(x)
sin(2x) = ?
sin(2x) = 2sin(x)cos(x)
cos^2(x) + sin^2(x) = ?
cos^2(x) + sin^2(x) = 1
What is the Real Fourier Sum (RFS)?
f(x) = a0/2 + Ξ£ a(n) cos(nx) + Ξ£ b(n) sin(nx)
What is a0?
a0 = 1/Ο β«(Ο, - Ο) f(x) dx
What is a(n)?
a(n) = 1/Ο β«(Ο, - Ο) f(x) cos(nx) dx
What is b(n)?
b(n) = 1/Ο β«(Ο, - Ο) f(x) sin(nx) dx
sin(Ο) = ?
sin(Ο) = 0
cos(Ο) = ?
cos(Ο) = -1
sin(0) = ?
sin(0) = 0
cos(0) = ?
cos(0) = 1
cos(-Ο) = ?
cos(-Ο) = -1
sin(-Ο) = ?
sin(-Ο) = 0
cos(2Ο) = ?
cos(2Ο) = 1
Eulerβs Formula
What is e^(inx) equal to?
e^(inx) = cos(nx) + i sin(nx)
Eulerβs Formula
What is e^(-inx) equal to?
e^(-inx) = cos(nx) - i sin(nx)
What is e^(-β)?
e^(-β) = 0
What is e^(0)?
e^(0) = 1
sin(2Ο) = ?
sin(2Ο) = 0
cos(Ο/2) = ?
cos(Ο/2) = 0
sin(Ο/2) = ?
sin(Ο/2) = 1
What is the diffusion equation?
β(^2) u = ( 1/Ξ±^2 ) ( βu / βt )
What is the laplace equation?
β^2 T = 0
Where T is a scalar function.
Laplace Equation
What are the general solutions when the plate is in the y-direction?
When the plate is in the y-direction,
T(x, y) = { e^ky } { cos(kx) }
________{ e^-ky } { sin(kx) }
Laplace Equation
What are the general solutions when the plate is in the x-direction?
When the plate is in the x-direction,
T(x, y) = { e^kx } { cos(ky) }
________{ e^-kx } { sin(ky) }
What are the general solutions to the diffusion equation?
u(x, t) = { cos(kx) } { e^(-k^2) (Ξ±^2) t }
{ sin(kx) }
Diffusion Equation
Give the boundary condition at x = 0 for an insulated bar.
βu/βx|x=0 = 0
Diffusion Equation
Give the boundary condition for x = L for an insulated bar.
βu/βx |x=L = 0
Diffusion Equation
Give the condition for k for an insulated bar.
Select cos(kx) solutions
with the condition that
kL = nΟ
Diffusion Equation
Give the boundary condition at x = 0 for a bar held at 0 degrees for t > 0.
u(0, t) = 0
Diffusion Equation
Give the boundary condition at x = L for a bar held at 0 degrees for t > 0.
u(L, t) = 0
Diffusion Equation
Give the condition for k for a bar held at 0 degrees at both edges.
Select sin(kx) solutions
with the condition that
kL = nΟ
What is the wave equation?
β^2 Ο = ( 1 / c^2 ) ( β^2 Ο / β t^2 )
What are the general solutions to the wave equation for y(x, t)?
y(x, t) = { cos(kx) } {cos(kct) }
________{ sin(kx) } { sin(kct) }
Wave Equation
What are the boundary conditions for a stretched string clamped at x = 0 and x = L?
y(0, t) = 0
y(L, t) = 0
Wave Equation
What are the boundary conditions for a tube of length L open at both ends?
βy(0, t) / βx = 0
βy(L, t) / βx = 0
Wave Equation
What are the boundary conditions for a tube of length L open at x = 0 and closed at x = L?
βy(0, t) / βx = 0
y(L, t) = 0
Wave Equation
What are the boundary conditions for a tube of length L closed at x = 0 and open at x = L?
y(0 , t) = 0
βy(L, t) / βx = 0
Wave Equation
Which general solutions are selected for a stretched string clamped at x = 0 and x = L?
sin(kx)
Wave Equation
Which general solutions are selected for a tube of length L open at both ends?
cos(kx)
Wave Equation
Which general solutions are selected for a tube of length L open at x = 0 and closed at x = L?
cos(kx)
Wave Equation
Which general solutions are selected for a tube of length L closed at x = 0 and open at x = L?
sin(kx)
Wave Equation
What is the condition for k when a stretched string is clamped at x = 0 and x = L?
kL = nΟ
Wave Equation
What is the condition for k when a tube of length L is open at both ends?
kL = nΟ
Wave Equation
What is the condition for k when a tube of length L is open at x = 0 and closed at x = L?
kL = (n + 1/2) Ο
Wave Equation
What is the condition for k when a tube of length L is closed at x = 0 and open at x = L?
kL = (n + 1/2) Ο
When finding the volume integral for spherical coordinates, what are the limits for r?
Upper: r = a
Lower: r = 0
When finding the volume integral for spherical coordinates, what are the limits for ΞΈ?
Upper: ΞΈ = Ο
Lower: ΞΈ = 0
When finding the volume integral for spherical coordinates, what are the limits for Ο?
Upper: Ο = 2Ο
Lower: Ο = 0
What is the general equation for the volume integral?
πΏπ = β1 β2 β3 πΏπ₯1 πΏπ₯2 πΏπ₯3
What is the equation for the volume integral for spherical polar coordinates?
πΏπ = π^2 sin π πΏπ πΏπ πΏΟ
What is the equation for the volume integral for cylindrical polar coordinates?
πΏπ = π πΏπ πΏπ πΏz
What is the equation for the volume integral for cartesian coordinates?
πΏπ = πΏπ₯ πΏπ¦ πΏz
What is C(0)?
1/2Ο β«(Ο, - Ο) f(x) dx
What is C(n)?
1/2Ο β«(Ο, - Ο) f(x) e^(-inx) dx
