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This rotational invariance suggests that the two-dimensional Laplacian should take a particularly...
Convert the polar equation to rectangular form and sketch its graph. r = 7 cot(0) csc(O)...
Convert the polar equation to rectangular form and sketch its graph. r = 7 cot(0) csc(O) Step 1 The polar coordinates (r, e) of a point are related to the rectangular coordinates (x, y) of the point as follows. x=rcos(0) cos y = r sin(0) sin e Step 2 The given polar equation can be rewritten as follows. r 7 cote csco 1 r = 7 coto sino 2 sin(0) = 7 coto Converting to rectangular coordinates using x =...
Please Help! If possible show every step my biggest issue is understanding how to get rid...
Please Help! If possible show every step my biggest issue is
understanding how to get rid of the Bessel Function. Thank you!
We've seen that the wave equation takes the form 1 af 2 12 = 0, where V2 is the Laplacian, v is the wave velocity, and f is the function describing the wave. In two Cartesian dimensions the Laplacian reduces to V2 f = 0f + 3f, which is a very convenient form for solving systems with rectangular...
P chem ll The kinetic energy T of a particle moving in three dimensions is given...
P chem ll
The kinetic energy T of a particle moving in three dimensions is given by im where i- dx / dt...etc and p is the magnitude of the linear momentum. Verify that this is true and show that in polar coordinates Where 9 -d9 / dt...etc. dx ах x =-= sin θ cosor + r cos θ cos φθ-r sin θ sin φφ. Similarly get y .from y = r sin θ sino dt and z from z-r...
Suppose that a scalar field is constant on a surface As shown in the lectures. there are two methods that one might use to obtain the normal to the surface, and they give the same direction (a) Let r...
Suppose that a scalar field is constant on a surface As shown in the lectures. there are two methods that one might use to obtain the normal to the surface, and they give the same direction (a) Let r(u, v) be a parametric form for the surface S. Use the vector identity to show that Our ar-λ▽u where λ is a scalar field. [Note: no marks will be awarded for simply stating that a term is zero. If it is...
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacia...
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacian is just like the Laplacian for the cylinder, but with the removed part เอ The structure of the Laplacian is what we call separable because the r and 0 terms are separate this allows us to solve certain physics problems on the disc by searching for solutions of the form f(r,0)-ar)b() The vibration of a circular drum head is described by 02t where u...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force pr...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy ,...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. p...
Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice for integrating over disks. Once we choose a coordinate system we must figure out the area form (dA) for that system. For example, when switching from rectangular to polar coordinates we must change the form of the area element from drdy to rdrd0. To determine that rdrde is the correct formula how the edges of...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addi...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addition, show the unit vectors r, θ, and φ at that point B. Write the vectors ŕ.0, and ф in terms of the unit vectors x, y, and г. Here's the easy way to do this 1. For r, simply use the fact that/r 2. For φ, use the following formula sin θ Explain why the above formula works 3. Compute θ via θ...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
Convert the polar equation to rectangular form and sketch its graph. r = 7 cot(0) csc(O) Step 1 The polar coordinates (r, e) of a point are related to the rectangular coordinates (x, y) of the point as follows. x=rcos(0) cos y = r sin(0) sin e Step 2 The given polar equation can be rewritten as follows. r 7 cote csco 1 r = 7 coto sino 2 sin(0) = 7 coto Converting to rectangular coordinates using x =...
Please Help! If possible show every step my biggest issue is
understanding how to get rid of the Bessel Function. Thank you!
We've seen that the wave equation takes the form 1 af 2 12 = 0, where V2 is the Laplacian, v is the wave velocity, and f is the function describing the wave. In two Cartesian dimensions the Laplacian reduces to V2 f = 0f + 3f, which is a very convenient form for solving systems with rectangular...
P chem ll
The kinetic energy T of a particle moving in three dimensions is given by im where i- dx / dt...etc and p is the magnitude of the linear momentum. Verify that this is true and show that in polar coordinates Where 9 -d9 / dt...etc. dx ах x =-= sin θ cosor + r cos θ cos φθ-r sin θ sin φφ. Similarly get y .from y = r sin θ sino dt and z from z-r...
Suppose that a scalar field is constant on a surface As shown in the lectures. there are two methods that one might use to obtain the normal to the surface, and they give the same direction (a) Let r(u, v) be a parametric form for the surface S. Use the vector identity to show that Our ar-λ▽u where λ is a scalar field. [Note: no marks will be awarded for simply stating that a term is zero. If it is...
(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacian is just like the Laplacian for the cylinder, but with the removed part เอ The structure of the Laplacian is what we call separable because the r and 0 terms are separate this allows us to solve certain physics problems on the disc by searching for solutions of the form f(r,0)-ar)b() The vibration of a circular drum head is described by 02t where u...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice for integrating over disks. Once we choose a coordinate system we must figure out the area form (dA) for that system. For example, when switching from rectangular to polar coordinates we must change the form of the area element from drdy to rdrd0. To determine that rdrde is the correct formula how the edges of...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addition, show the unit vectors r, θ, and φ at that point B. Write the vectors ŕ.0, and ф in terms of the unit vectors x, y, and г. Here's the easy way to do this 1. For r, simply use the fact that/r 2. For φ, use the following formula sin θ Explain why the above formula works 3. Compute θ via θ...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
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