Sonata form (also sonata-allegro form or first movement form) is a musical structure consisting of three main sections: an exposition, a development, and a recapitulation. It has been used widely since the middle of the 18th century (the early Classical period).
While it is typically used in the first movement of multi-movement pieces, it is sometimes used in subsequent movements as well—particularly the final movement. The teaching of sonata form in music theory rests on a standard definition and a series of hypotheses about the underlying reasons for the durability and variety of the form—a definition that arose in the second quarter of the 19th century. There is little disagreement that on the largest level, the form consists of three main sections: an exposition, a development, and a recapitulation; however, beneath this general structure, sonata form is difficult to pin down to a single model.
The standard definition focuses on the thematic and harmonic organization of tonal materials that are presented in an exposition, elaborated and contrasted in a development and then resolved harmonically and thematically in a recapitulation. In addition, the standard definition recognizes that an introduction and a coda may be present. Each of the sections is often further divided or characterized by the particular means by which it accomplishes its function in the form.
After its establishment, the sonata form became the most common form in the first movement of works entitled "sonata", as well as other long works of classical music, including the symphony, concerto, string quartet, and so on. Accordingly, there is a large body of theory on what unifies and distinguishes practice in the sonata form, both within and between eras. Even works that do not adhere to the standard description of a sonata form often present analogous structures or can be analyzed as elaborations or expansions of the standard description of sonata form.
I am given the following set of 4x4 matrices. How can i justify that they form a basis for the Lie Algebra of the group SO(4)? I know that they must be real matrices, and AA^{T}=\mathbb{I}, and the detA = +-1. Do i show that the matrices are linearly independent, verify these properties, and...
Homework Statement
For the following integral, find F and its partial derivatives and plug them into the Euler Lagrange equation
$$F(y,x,x')=y\sqrt{1+x'^2}\\$$
Homework Equations
Euler Lagrange equation : $$\frac{dF}{dx}-\frac{d}{dy}\frac{dF}{dx'}=0$$
The Attempt at a Solution...
hello this is my first topic here and i hope good discussion or answer to my question
As i understand the Maxwell equation keep its form in all frames so why i need to make a covarient formulation form of electrodynamics ?
for example what the covarient form of continuity equation give me !
Please see attached.
I am trying to show that
## T_{p} f (\tau + 1) = T_{p} f (\tau ) ##
##f(\tau) \in M_k ## and so can be written as a expansion as ##f(\tau)=\sum\limits^{\infty}_{0}a_{n}e^{2 \pi i n \tau } ##
##f(\tau + 1) = f(\tau) ## since ##e^{2\pi i n} = 1##
Similarly ##f(p\tau + p) =...
So I always thought that geometry is somewhat different from the rest of math. I mean, most of math is about numbers and relations. While geometry is about space.
Does analysis connect the two? For example, the hypotenuse of a triangle is just a truncated portion of the number line that has...
Problem:
Express the quadratic form:
q=x1x2+x1x3+x2x3
in canonical form using Lagrange's Method/Algorithm
Attempt:
Not really applicable in this case due to the nature of my question
The answer is as follows:
Using the change of variables:
x1=y1+y2
x2=y1-y2
x3=y3
Indeed you get...
Hello! This is my first post on these forums.
So I was stuck with this question in my Mathematical Analysis exam, and it is as follows:
ƒ(x) = 0 if x ∉ ℚ and (p + π) / (q + π) - (p / q) if x = (p / q) ∈ ℚ (reduced form).
1- Prove ƒ is discontinuous at all rational numbers except 1:
This is...
##\theta(\tau, A) = \sum\limits_{\vec{x}\in Z^{m}} e^{\pi i A[x] \tau } ##
##=\sum\limits^{\infty}_{n=0} r_{A}(n)q^{n} ##,
where ## r_{A} = No. [ \vec{x} \in Z^{m} ; A[\vec{x}] =n]##
where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive...
Homework Statement
Let's say we have a mass of 5kg at a height of 3 m so it's potential energy is mgh = 147J/1.6e-19 = 9.19 e20 eV. Now we know that E = mc^2... so when finding the mass of this potential energy we get 10.2e3 kg. What the hell is that supposed to mean?
Homework Equations
None...
$\tiny{s6.12.25}$
$\textsf{If $v$ lies in the first quarter and makes an angle }\\$
$\textsf{$\pi/3$ with the positive x-axis and $\left| v \right|$=4} $
$\textsf{find $v$ in component form.}$
\begin{align}
\displaystyle
v&=\langle 2\sqrt{3},2\rangle \\
\end{align}
this is probably correct...
dy/dx = xe^(y-2x) , i am asked to form differential equation using this equation . the ans given is (e^-y) = 0.5(e^-2x)(x+0.5) + a , how to get the answer? btw , i have attached my working
Dear All
It is known that pontryagin densities are defined in even dimension space, let's say i am concerned with 4 dim space time. We also have a certain group G. What is the formula of pontryagin densities for arbitrary group? Larger group?
The volume form on the unit sphere ##S^{n}## in ##\mathbb{R}^{n+1}## is given by
$$i_{{\bf r}}\ dx^{1}\wedge \dots \wedge dx^{n+1}=\sum (-1)^{i-1}x^{i}dx^{1}\wedge\dots \widehat{dx^{i}} \dots \wedge dx^{n+1}.$$
Why must the volume form ##dx^{1}\wedge \dots \wedge dx^{n+1}## act on the vector...
I am pretty confused about how to write Navier-Stokes Equation into conservation form, it seems that from my notes,
first, the density term with the pressure gradient dropped out.
and second, du^2/dx seems to be equal to udu/dx.
Why is it so? I attached my notes here for your reference.
http://www.physicspages.com/2014/08/22/electromagnetic-waves-in-vacuum/
Sorry
i have no idea how to get the last step in the vector form
how to convert and mix it?thank you
Homework Statement
Reduce ##xy+zy## to diagonal form.
Homework Equations
The desired diagonal form is ##Q(\vec{x})=(\alpha_1(\vec{x}))^2+...+(\alpha_k(\vec{x}))^2-(\alpha_{k+1}(\vec{x}))^2-...-(\alpha_{k+l}(\vec{x}))^2,## where ##\alpha_i## are linearly independent linear functions. Also known...
Homework Statement
Express the complex number (−3 +4i)3 in the form a + bi
Homework Equations
z = r(cos(θ) + isin(θ))
The Attempt at a Solution
z = -3 + 4i
z3 = r3(cos(3θ) + isin(3θ))
r = sqrt ((-3)2 + 42)
= 5
θ = arcsin(4/5) = 0.9273
∴ z3 = 53(cos(3⋅0.9273) + isin(3⋅0.9273))
a = -117
b...
Homework Statement
Prove that ##[L_i,x_j]=i\hbar \epsilon_{ijk}x_k \quad (i, j, k = 1, 2, 3)## where ##L_1=L_x##, ##L_2=L_y## and ##L_3=L_z## and ##x_1=x##, ##x_2=y## and ##x_3=z##.
Homework Equations
There aren't any given except those in the problem, however I assume we use...
Mod note: Based on an attachment in a later post in this thread, the actual expression is
##(9 - 16\cos^2(\theta))^{3/2}##
Homework Statement
: https://www.physicsforums.com/posts/5610105/
Homework EquationsThe Attempt at a Solution
my working is (9^1.5) - (16^1.5)[ (cos theta)^1.5 ] =...
In particular how does matter "clump" together to form stars and planets, and how do Galaxy/star systems form?
For the latter question is the answer simply that near massive enough bodies, the spacetime curvature is significant enough that the geodesics within its vicinity are closed curves...
Homework Statement
in this question , solving equation 1 and 2 , how to get the
F1 = (A1E1 P ) / ( 2A1E1 + A2E2) ? and also F2 ?
Homework EquationsThe Attempt at a Solution
we know that F2 = P -2F1 )
I have tried to sub (F2 / A2E2 ) in F1 = (A1E1 P ) / ( 2A1E1 + A2E2) , but i ended up...
Homework Statement
I have the second order diff eq:
Solving by Laplace transform gets me to:
I could use the inverse laplace transform that takes me back to e^{at}cos(bt) with b=0, but that only solves for the homogeneous (complementary) part of the equation, it won't reproduce the dirac...
The material I am studying express the Ricci form as
##R = i{R_{\mu \bar \nu }}d{z^\mu } \wedge d{{\bar z}^\nu } = i\partial \partial \log G##
where ##G## is the determinant of metric tensor, but I am not sure what does ##\log G## here, can anybody help?
Hi.
Is the Maxwell equation
$$\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon_0}$$
even true in the stronger form
$$\frac{\partial E_i}{\partial x_i}=\frac{\rho}{3\cdot\varepsilon_0}\enspace ?$$
I guess not, since I haven't found a source suggesting this. But shouldn't the isotropic electric field...
Okay, I'm following a series of video lectures on applications of finite element method to engineering, and the tutor started by demonstrating the mathematical background of FEM using a simple heat transfer problem. He derived the governing equation (in just one dimension):
(1)...
$-x^2+4x-1$ should be converted to the vertex form of $y=k-(x-h)^2$
How can this be solved by factoring or any other method ?
My attempt to solve this problem , I will be using the completing the square method,
$\left(-x^2+4x+\frac{-b}{2a}\right)=1+\frac{-b}{2a}$
Here $\frac{-4}{-2}=2$...
Hi, I'm a high school science teacher. Most textbooks classify EM radiation as kinetic energy. But this doesn't seem right to me. As a photon is massless it's hard to see how it can have kinetic energy which is 1/2 mv^2.
It could be said that it has energy hf and therefore mass hf/c^2. Then its...
If equation of motion(K-G Eqn.,) follows,
∂μ∂μΦ+m2Φ=ρ
where 'ρ' is point source at origin.
How time independent form of above will become,
(∇2-m2)Φ(x)=gδ3(x)
where g is the coupling constant,
δ3(x) is three dimensional dirac delta function.
I was looking for a derivation of E=(2kλ)/r for an infinite line of charge. I understood that you need to use Gauss's Law and a cylinder around the line. When looked it up, I found this: http://www.vizitsolutions.com/portfolio/gausslaw/lineCharge.html
He starts out with ∫E⋅dA=4πq. I have never...
Hello all,
I am trying to bring this:
(p \iff q ) \implies r
into a CNF form. I have started with the logical equivalences:
(p \implies q) = \lnot p\lor q
(p \iff q) = (p \land q)\lor (\lnot p \land \lnot q)
and then I have applied De Morgan's rules and the distribution rules, but...
Homework Statement
Homework Equations
Cartesian to Cylindrial
The Attempt at a Solution
What I was doing is that, I changed the limits of z, and the function.
Homework Statement
Show that the Lorentz Force Law, \frac{dp^{\nu}}{d \tau} = -q U_{\mu} F^{\mu \nu}, is consistent with P^\mu P_\mu= -m^2. Here U is the 4-velocity, F is the Electromagnetic Tensor, and p is the 4-momentum. (Minkowski Space)
Homework Equations
As stated above.
The Attempt at...
I have been around Sailboats all my life. For the last 25 years I have built and raced r/c model sailboats.
My current question, I hope, can be more of a discussion than just a simple question and answer. Since I am into the design of all aspects of boat and sails. I have a question about...
Simplify the function f(x)= (x-(1+i))(x-(1-i)) .
Is the function you wrote in factored form [ f(x)= (x-(1+i))(x-(1-i)) ] , the same as the original function in
standard form [ f(x) = x^2 - 2x +2) ] ?
Homework Statement
Express F1 and F2 in cartesian vector form
Homework EquationsThe Attempt at a Solution
I feel fairly confident in my work for F2 but F1 not so much. Especially the j component of F1. I assume my F(1x) component runs from the z axis to the tip of F1 and my F(1k) component...
Hi PF!
I was reading my book and I understand the following $$\sum \vec{F} = \frac{\partial}{\partial t} \iiint_{CV} \rho \vec{u} dV +\iint_{CS} \rho \vec{u} ( \vec{u_{rel}} \cdot \hat{n}) dS$$ ##CV## is a control volume, ##CS## is control surface, ##u## is velocity, ##u_{rel}## is velocity...
Hi
For brevity one usually writes Fredholm integral equation of the 2nd kind
$\psi(x)=f(x)+\int_{a}^{b} \,k(x,s)\psi(s) ds$
into the form
$\psi=f+K \psi$
where $K$ is the operator kernel
My question can one write an integro differential equation
$\d{\psi(x)}{x}=f(x)+\int_{a}^{b}...
Homework Statement
Suppose that a surface has an equation in cylindrical coordinates of the form ##z=f(r)##. Explain why it must be a surface of revolution.
Homework EquationsThe Attempt at a Solution
I consider ##z=f(r)## in terms of spherical coordinates.
## p cosφ = f \sqrt{(p sinφcosθ)^2...
Hello,
I have a maybe unusual question. In a paper, I recently found the equation $$\mathcal{L}_v(v_i dx^i) = (v^j \partial_j v_i + v_j \partial_i v^j) dx^i$$
Where v denotes velocity, x spatial coordinates and \mathcal{L}_v the Lie derivative with respect to v. Now I'm an undergraduate who...
Homework Statement
Given x(t)=8cos(70\pi t)+4sin(132\pi t)+8cos(24\pi t), find the Fourier transform X(f) in the form of \delta function.
Homework Equations
X(f)=\int ^{\infty}_{-\infty}x(t)e^{-j\omega _0t}dt
cos(\omega t)=\frac{e^{j\omega t}+e^{-j\omega t}}{2}
sin(\omega t)=\frac{e^{j\omega...
I am trying to establish a Rationalist approach to Physics as a side project, and have taken Hamilton's Principle as one of the few postulates in my work. I've developed the concept enough to arrive at the usual stuff, like the Euler-Lagrange equations, Newton's First Law and Nöther's Theorem...
http://arxiv.org/pdf/1609.00716.pdf
Date:
September 2, 2016
Source:
European Space Agency (ESA)
Summary:
ESA's Planck satellite has revealed that the first stars in the Universe started forming later than previous observations of the Cosmic Microwave Background indicated. This new analysis also...
Homework Statement
Write this complex number in the form "a+bi"
a) cos(-pi/3) + i*sin(-pi/3)
b) 2√2(cos(-5pi/6)+i*sin(-5pi/6))
Homework Equations
my only problem is that I am getting + instead of - on the cosinus side.(real number)
The Attempt at a Solution
a) pi/3 in the unit circle is 1/2...
Homework Statement
Homework Equations
Theta = arctan (y/x)
The Attempt at a Solution
Hopefully this is the right section to post in, but I am a bit confused with complex numbers. I am working on the problems above and I just wanted to make sure I am doing each part correctly. I think A...
Homework Statement
Write the given numbers in the polar form ##re^{i\theta}##.
## \frac {2i} {(3e^{4+i})} ##
Homework Equations
## z = re^(i\theta) ##
## \theta = Arg(z) ##
## r = |z| = \sqrt { x^2 + y^2 } ##
The Attempt at a Solution
I'm not really sure how to go about the exponential...
Hey all, I need the complex version of the sigmoid function in standard form, that is to say $$f(\alpha) =\frac{1}{1+e^{-\alpha}} , \hspace{2mm}\alpha = a+bi , \hspace{2mm} \mathbb{C} \to \mathbb{C}$$ in the simplified form: $$f = m+ni$$ but found this challenging, for some reason i assumed...
Hello! (Wave)
I want to write the language of the automaton with the following transition function in regular form with $A$ as an initial state and $B,D$ as final states.
$$\delta:\begin{matrix}
& & 0 & 1\\
& A & B & C\\
& B & C & D\\
& C & D & B\\
& D & D & C
\end{matrix}$$
I have...