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.
one book example
$\textsf{Suppose that A is a matrix whose characteristic polynomial is}$
$(\lambda-2)^2(\lambda + 1)^2,
\quad \dim\left(E_2\right)=1
\quad \dim\left(E_{-1}\right)=2$
$\textsf{Find the Jordan Normal Form of A Find the Jordan Normal Form of A}$
$$\quad \dim\left(E_2\right)=1...
Homework Statement
I learned that Covalent Bonds form between different specific atoms ( with similar electro-negativity ) with electrons.
However, I wondered what type of bond would form between the different atoms if they had no electrons?
Also , if I have 2 atoms with similar...
Do the Maxwell equations in the usual 3-vector form have the same form in any Lorentz frame? For example, the one that says ##\nabla \cdot \vec B = 0## will be valid in another, primed Lorentz frame? That is ##\nabla' \cdot \vec {B'} = 0##?
Homework Statement
I want to manipulate an equation to suit a desired form.
Homework Equations
##(-h^2/2uc)*(dp/dx)*ln((1+c*(x/h)^2)/(1+c))##
becomes
##-(h^2/2u)*(dp/dx)*(1-(x/h)^2)##
The Attempt at a Solution
I have no idea, I'm not even sure how the natural log disappears. [/B]
Homework Statement
ABC is an isosceles triangle such that
AB = AC
A has coordinates (4, 37)
B and C lie on the line with equation 3y = 2x + 12
Find an equation of the line of symmetry of triangle ABC.
Give your answer in the form px + qy = r where p, q and r are integers. Show clear...
The Biot-Savart law which describes a magnetic field created by a displacement current: $$\frac{dB}{dV}=\frac{\mu_0\epsilon_0}{4\pi}\frac{\frac{∂E}{∂t}×r}{r^2}$$
What's the relativistically co-variant form of this equation?
Is the introduction of speed of light propagation delays enough, or...
I'm searching, but so far I have not found a derivation of the coordinates shown by wikipedia in the very beggining of https://en.wikipedia.org/wiki/Rindler_coordinates#Characteristics_of_the_Rindler_frame.
It seems obvious from the relation ##X^2 - T^2 = 1 / a^2##, (##c = 1##), that ##X =...
Hi! I have a simple set of nonlinear equations
1) 3x = 30
2) x+2y = 20
3) x + y*z = 15
Clearly the solution to this is (10,5,1) but I want to find a robust way to solve this type of problem [A]x=b (where [A] is a simple function of x) which doesn't involve numerically solving using Newtons...
Homework Statement
If the current flow, in a branch of a circuit, is a function of say (√(x + 2)-2)/(x-2) (or any such that give an indeterminate form at a certain value) of an input source current x.
What current will be flowing on that part of the circuit, when the function become...
##
\newcommand{\ihat}{\hat{\boldsymbol{\imath}}}
\newcommand{\jhat}{\hat{\boldsymbol{\jmath}}}
\newcommand{\khat}{\hat{\boldsymbol{k}}}
##
Several times now I've seen the following technique for deriving the component form of the dot product. It always felt clean and simple until last night when...
I don't fully understand how to work out the impedance from the given equation (5j-5)x(11j-11)/(5j-5)+(11j-11). Any help would be greatly appreciated. Thanks.
The answer needs to be in rectangular and polar form.
This is no homework. I have come across a conjecture in a book called The art of the infinite:the pleasures of mathematics. I want to understand how to prove it.
Homework Statement
Consider a 3-rhythm starting with 2: ## 2, 5, 8, 11, 14, 17...##
The each number in this sequenc has the form...
Homework Statement
Given: sin(Πx/a)e6Πix/Na
and e2Πi/a(7/N+4)x
can these equations be represented in Bloch form?[/B]
Homework Equations
Given that Bloch form can be represented as:
Ψ(x) = u(x) eikx[/B]
The Attempt at a Solution
sin(Πx/a)eikx w/n = 3
and...
this is what is given
so by addition
$$\begin{bmatrix}x_1\\y_1\\5z_1\end{bmatrix}
\oplus
\begin{bmatrix} x_2\\y_2\\5z_2
\end{bmatrix}
=
\begin{bmatrix}
x_1+x_2\\y_1+y_2\\5z_1+5z_2
\end{bmatrix}
=
\begin{bmatrix}
X\\Y\\10Z
\end{bmatrix}$$
uhmmmm really?
Determine if the set of vectors
$\begin{bmatrix}
x\\y\\3x+2y
\end{bmatrix}$ $\in \Bbb{R}^3$
form a vector space
(with the usual addition and scalar multiplication for vectors in $\Bbb{R}^3$).OK first of all this doesn't have z in it.
So I don't know if this meets the requirement of...
Homework Statement
This problem is from V.I Arnold's book Mathematics of Classical Mechanics.
Q) Show that every differential 1-form on line is differential of some function
Homework Equations
The differential of any function is
$$df_{x}(\psi): TM_{x} \rightarrow R$$
The Attempt at a Solution...
Homework Statement
Write ##5-3i## in the polar form ##re^\left(i\theta\right)##.
Homework Equations
$$
|z|=\sqrt {a^2+b^2}
$$
The Attempt at a Solution
First I've found the absolute value of ##z##:
$$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$.
Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...
The Role of H in the quadratic function ( vertex form)
i get that this is how its written on a graph y=(x-2)^2+k
that the graph looks as if the value of h is positive as in +2 ( however its value is actually negative)
looks like it shifted right my textbook contradicts itself
y=3(x-1)^2 +2...
Homework Statement
We are asked to show that:
## \frac{d^2x_\mu}{d\tau^2}= \frac{1}{2} \frac{dx^\nu}{d\tau} \frac{dx^{\rho}}{d\tau} \frac{\partial g_{\rho \nu}}{\partial x^{\mu}} ##
( please ignore the image in this section i cannot remove it for some reason )
Homework Equations
The...
Homework Statement
Suppose that a smooth differential ##n-1##-form ##\omega## on ##\mathbb{R}^n## is ##0## outside of a ball of radius ##R##. Show that $$
\int_{\mathbb{R}^n} d\omega = 0.
$$
Homework Equations
[/B]
$$\oint_{\partial K} \omega = \int_K d\omega$$
The Attempt at a Solution...
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
I am trying to complete the last part of this question, part 5(c). My professor has told me that the form factor $$F(q)\rightarrow1$$ as $$q\rightarrow0$$ but I am unsure how to show this.
I believe that $$\lim_{{q...
I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll and he writes:
Quote: A useful characterisation of the metric is obtained by putting ##g_{\mu\nu}## into its canonical form. In this form the metric components become $$ g_{\mu\nu} = \rm{diag} (-1...
Homework Statement
A vector n of magnitude 8 units is inclined to x,y and z axis at 45, 60 and 60 degrees resoectively.If the plane passes through (root2, -1, 1) and is normal to n then find its equation.
Homework Equations
(r-a).n=0 where r is position vector of a point on plane, a is a point...
Homework Statement
lim x--->0 |x|^sinx is?
Homework Equations
lim x-->0 f(x)^g(x), if both functions tend to 0, limit is equal to e^log[f(x).g(x)] with the same limit..(i)
The Attempt at a Solution
when x>0, it is x^sinx and x<0 it is -1/x^sinx. putting the first case in (i) we get...
Homework Statement
Given the above lambda system, is it wrong to say that the density matrix is of the form ## \rho = c_1|1> + c_2|2> + c_3|3> ## ? Hence when written in matrix form (basis of ##|i>##), ## \rho ## is a diagonal matrix who's elements are the ##c_i##s?
Hey! :o
Let $V$ be a vector space with with a 5-element basis $B=\{b_1, \ldots , b_5\}$ and let $v_1:=b_1+b_2$, $v_2:=b_2+b_4$ and $\displaystyle{v_3:=\sum_{i=1}^5(-1)^ib_i}$.
I want to determine all subsets of $B\cup \{v_1, v_2, v_3\}$ that form a basis of $V$.
Are the desired subsets the...
In ##c=1## units, from my SR courses I was told for example, that the Minkowski metric ## ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 ## can be written in matrix form as the below..
\eta =
\begin{pmatrix}
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
And it was just...
Hey.
I have the following question to solve:
* Given a matrix A that is size m x n and m>n.
Let R be the RREF that we get by Gaussian elimination of A.
Prove that the system equation Ax=0 has only one solution iff in every column of R there is a leading element.
I have some answer of...
Hey! :o
Let $V$ be a vector space. Let $b_1, \ldots , b_n\in V$ and let $\displaystyle{b_k':=\sum_{i=1}^kb_i}$ for $k=1, \ldots , n$.
I want to show that $\{b_1, \ldots , b_n\}$ is a basis of $V$ iff $\{b_1', \ldots , b_n'\}$ is a basis of $V$. I have done the following:
Let $B:=\{b_1...
Homework Statement
We are given a Lorentz four-vector in "isospin space" with three components ##\vec v^{\mu} = (v^{\mu}_1, v^{\mu}_2, v^{\mu}_3)## and want to express the covariant derivative $$D^{\mu} = {\partial}^{\mu} - ig\frac {\vec \tau} {2}\cdot \vec v^{\mu}$$ explicitly in ##2\times 2##...
Homework Statement
In picture, first-order reflection from the reflection planes shown occurs when an x-ray beam of wavelength ##0.260 nm## makes an angle ##\theta=63.8°## with the top face of the crystal. What is the unit cell size ##a_0##?
Homework Equations
Bragg law
$$d=\frac{ n...
In many texts I have seen, Gauss theorem has the form of$$\frac{q}{\epsilon_0}=\oint\vec{E}d\vec{A}$$
Why a line integral symbol was used for this surface integral everywhere? The more I see it the more I believe there is something wrong with my understanding about this.
I didn't think too much...
Find the closed form (or) analytic expression form for the following integral
$$
\hspace{0.3cm} \large {\int_{0} ^{\infty} \frac{\frac{1}{x^4} \hspace{0.1cm} e^{- \frac{r}{x^2}}\hspace{0.1cm}e^{- \frac{r}{z^2}} }{ \frac{1}{x^2} \hspace{0.1cm} e^{- \frac{r}{x^2}}+ \frac{1}{y^2}...
I have received (unasked) a digital edition of "Laws of Form" (1969) by G. Spencer-Brown; I have glanced at it, and also at the Wikipedia article https://en.wikipedia.org/wiki/Laws_of_Form. OK, another logical system; logical journals (e.g. by ASL) are full of them, and I am not sure whether...
Homework Statement
Show that ##\displaystyle \bigcup_{n=2}^\infty \left[ \frac{1}{n} , \frac{n}{n+1} \right] = (0,1)##.
Homework EquationsThe Attempt at a Solution
I'm not sure how to show this rigorously. It is sufficient to note that ##\lim_{n\to\infty} \frac{1}{n} = 0## and that...
So the question is:
express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form
I know r=1 and tangent=pi/4
Using the theorem i get 1(cos (2pi) +i*sin (2pi)) which becomes 1(1*i)=1*i however WebAssign says this is incorrect. I've also tried "0+1i" and just "i"
What am I doing wrong?
Hi. After arranging the dynamic contact between a elastic ball against a flat, I have reached the following differential equation for the motion during the contact:
m·x’’+(k+c·x’)·x^n=0
with m,c,k>0 and for exponent n --> 1<n<2
Any functional form for this equation? I have solved it...
Convert the equation to polar form
8x=8y
I thought it would be
8*r*cos(theta)=8*r*sin(theta)
Said it was incorrect
then I thought I needed to divide by 8 to remove it, giving me:
r*cos(theta)=r*sin(theta)
But that was also incorrect and now I am stuck
Let's say we have a given matrix ##G##. I want to find a set of ##M## matrices so that ##MG = GM## and prove that this is a group. How can I approach this problem?
The problem statement
Express exp(3+π*i) in Cartesian Form.
The attempt at a solution
Equating
e^(3+πi) = e^(x)e^(iy) = e^(x)(cos(y) + isin(y))
then
e^(x)cos(y) = 3
e^(x)sin(y)=π
now
|e^(3+πi)| = e^(x)
so x = sqrt(9+π^2)
then
cos(y) = 3/sqrt(9+π^2)
sin(y) = π/sqrt(9+π^2)
at this point i don't...
nmh{796}
$\textsf{Suppose $Y_1$ and $Y_2$ form a basis for a 2-dimensional vector space $V$ .}\\$
$\textsf{Show that the vectors $Y_1+Y_2$ and $Y_1−Y_2$ are also a basis for $V$.}$
$$Y_1=\begin{bmatrix}a\\b\end{bmatrix}
\textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$
$\textit{ then }$...
Hey! :o
We have the sequence $$0, \ 2 , \ -6, \ 12, \ -20, \ \ldots$$ Its recursive definition is \begin{align*}&a_1=0 \\ &a_{n+1}=(-1)^{n+1}\cdot (a_n+2\cdot n)\end{align*} or not?
How can we convert that in the explicit form? (Wondering)