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1. Spherical waves. Starting with the wave equation VE in spherical polar ,,2 2.1 in spherical po...
In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical...
In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical symmetry about the polar axis is bounded on the polar axis can be expressed as u(r, 0) = Rm(r)P,(cos 0), (A) where P is the Legendre polyomial of degree n, and R(r) is the general solution of the differential equation *() - n(n + 1)R = 0, (r > 0), dr dr where n is a non-negative integer. (You are not asked to show...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r,...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
1. We'll use separation of variables to solve the Schrodinger equation in spherical geometry Show...
Quantum Physics
1. We'll use separation of variables to solve the Schrodinger equation in spherical geometry Show, that if the wave function takes the form 9(r,6, o) . R (r) (6)$(o) that the SchrodinDer equation can be separated in three equations d. (sin ) +1(1+1)sin2@62 ㎡Θ, and b) Show that imposing the boundary condition ф (ф)-ф (ф 2x) feguires that m-0, 1, 2, 3, ' . . dThe hrst few Legendre polymomials are given by 0-63-15 The "associated Legendre functions"...
a) By direct substitution determine which of the following functions satisfy the wave equation. 1. g(x,...
a) By direct substitution determine which of the following functions satisfy the wave equation. 1. g(x, t) = Acos(kx − t) where A, k, are positive constants. 2. h(x, t) = Ae where A, k, are positive constants. 3. p(x, t) = Asinh(kx − t) where A, k, are positive constants. 4. q(x, t) = Ae where A, a, are positive constants. 5. An arbitrary function: f(x, t) = f(kx−t) where k and are positive constants. (Hint: Be careful with...
(2, Consider the Laplace equation for a ball of radius R described in spherical coordinates (T,0) 2 1 +00n3 2 0 wher...
(2, Consider the Laplace equation for a ball of radius R described in spherical coordinates (T,0) 2 1 +00n3 2 0 where 0 is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w =cos e given by 12 F" (T) +2r F (r) - n(n + 1 ) Fr (r) = 0, (1 w2)G (w) - 2wG (w) + n(n+ 1)G, (w)...
Consider the Laplace equation for a ball of radius R described in spherical coordinates (r, 0) 2 1 cot 72 0= n where...
Consider the Laplace equation for a ball of radius R described in spherical coordinates (r, 0) 2 1 cot 72 0= n where is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w:= cos e given by 2 F" (r) +2rF (r) - n(n + 1) F, (r) = 0, (1 w2)G (w) - 2wG", (w) +n(n +1)G, (w) 0. (n 0,1,2,....
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3)....
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
solve for (c) ~ (g) especially tricky integration is need to be solved solve for (d) ~(g) (c) is solved 2. Using polar coordinates: (a) Show that the equation of the circle sketched is r 2a cos 0....
solve for (c) ~ (g)
especially tricky integration is need to be solved
solve for (d) ~(g)
(c) is solved
2. Using polar coordinates: (a) Show that the equation of the circle sketched is r 2a cos 0. Hint: Use the right triangle OPGQ (b) By integration, find the area of the distk P(r, e) 2a r < 2a cos θ Find the centroid of the area of the first quadrant (c) half disk. (d) Find the moments of inertia...
1. If a function f(x,y) has a local maximum then it is not necessary that it...
1. If a function f(x,y) has a local maximum then it is not necessary that it has also a local minimum True False 2. If a vector field F is conservative then we can not find a potential functions. True False 3. Suppose that P and Q have continuous first-order partial derivatives on a domain D and consider the vector field F = Pi+Qj. Then F is conservative if op 80 True False 4. If D is a rectangle, then...
Hello! I need help answering these Partial Differential Equations exercises! Exercise 1 Find the general solution...
Hello! I need help answering these Partial Differential
Equations exercises!
Exercise 1 Find the general solution of the cquation ury(r, y) 0 in terms of wo arbitrary functions. Exercise 2 Verify that 2c9(s)ds tcontinuously differentiable function. Hint: Here you will need to use iz' ution to the wave equation u2S, where c is a constant and g is 1's rule for differentiating an integral with respect to a parameter that a given urs n the limits of integration: b(t) F(b(t))b'...
In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical symmetry about the polar axis is bounded on the polar axis can be expressed as u(r, 0) = Rm(r)P,(cos 0), (A) where P is the Legendre polyomial of degree n, and R(r) is the general solution of the differential equation *() - n(n + 1)R = 0, (r > 0), dr dr where n is a non-negative integer. (You are not asked to show...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
Quantum Physics
1. We'll use separation of variables to solve the Schrodinger equation in spherical geometry Show, that if the wave function takes the form 9(r,6, o) . R (r) (6)$(o) that the SchrodinDer equation can be separated in three equations d. (sin ) +1(1+1)sin2@62 ㎡Θ, and b) Show that imposing the boundary condition ф (ф)-ф (ф 2x) feguires that m-0, 1, 2, 3, ' . . dThe hrst few Legendre polymomials are given by 0-63-15 The "associated Legendre functions"...
(2, Consider the Laplace equation for a ball of radius R described in spherical coordinates (T,0) 2 1 +00n3 2 0 where 0 is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w =cos e given by 12 F" (T) +2r F (r) - n(n + 1 ) Fr (r) = 0, (1 w2)G (w) - 2wG (w) + n(n+ 1)G, (w)...
Consider the Laplace equation for a ball of radius R described in spherical coordinates (r, 0) 2 1 cot 72 0= n where is the zenith angle and assume u is independent on the azirnuth angle d. a) By separation of variables, derive two ordinary differential equations of r and w:= cos e given by 2 F" (r) +2rF (r) - n(n + 1) F, (r) = 0, (1 w2)G (w) - 2wG", (w) +n(n +1)G, (w) 0. (n 0,1,2,....
BOX 5.1 The Polar Coordinate Basis Consider ordinary polar coordinates r and 0 (see figure 5.3). Note that the distance between two points with the same r coordinate but separated by an infinitesimal step do in 0 is r do (by the definition of angle). So there are (at least) two ways to define a basis vector for the direction (which we define to be tangent to the r = constant curve): (1) we could define a basis vector es...
solve for (c) ~ (g)
especially tricky integration is need to be solved
solve for (d) ~(g)
(c) is solved
2. Using polar coordinates: (a) Show that the equation of the circle sketched is r 2a cos 0. Hint: Use the right triangle OPGQ (b) By integration, find the area of the distk P(r, e) 2a r < 2a cos θ Find the centroid of the area of the first quadrant (c) half disk. (d) Find the moments of inertia...
1. If a function f(x,y) has a local maximum then it is not necessary that it has also a local minimum True False 2. If a vector field F is conservative then we can not find a potential functions. True False 3. Suppose that P and Q have continuous first-order partial derivatives on a domain D and consider the vector field F = Pi+Qj. Then F is conservative if op 80 True False 4. If D is a rectangle, then...
Hello! I need help answering these Partial Differential
Equations exercises!
Exercise 1 Find the general solution of the cquation ury(r, y) 0 in terms of wo arbitrary functions. Exercise 2 Verify that 2c9(s)ds tcontinuously differentiable function. Hint: Here you will need to use iz' ution to the wave equation u2S, where c is a constant and g is 1's rule for differentiating an integral with respect to a parameter that a given urs n the limits of integration: b(t) F(b(t))b'...
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