Form Definition and 1000 Threads

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.

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  1. karush

    MHB 25.3 Find the Jordan Normal Form of A

    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...
  2. C

    Can Covalent Bonds form between atoms with no electrons?

    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...
  3. K

    I Maxwell's Equations in Different Lorentz Frames: Are They Equivalent?

    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##?
  4. S

    How do I manipulate this to the form desired?

    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]
  5. N

    Find an equation of a line of symmetry in the form px+qy = r

    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...
  6. T

    I Relativistic form of the displacement current using the Biot-Savart Law

    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...
  7. K

    I Finding Most General Form of Rindler Coordinates

    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 =...
  8. M

    Solve simple nonlinear equations in the form [A]x=b

    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...
  9. navneet9431

    B Prism Forming Image: Nature & Types

    Do Prism form image? If they do form image, then what is the nature of the image formed?
  10. R

    When current flow reach indeterminate form

    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...
  11. ibkev

    I Dot product definition: deriving component form

    ## \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...
  12. M

    MHB Finding the impedance in rectangular and polar form

    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.
  13. Entertainment Unit

    Convergence of a series given in non-closed form

    Homework Statement Determine whether the given series is absolutely convergent, conditionally convergent, or divergent. ##\frac{1}{3} + \frac{1 \cdot 4}{3 \cdot 5} + \frac{1 \cdot 4 \cdot 7}{3 \cdot 5 \cdot 7} + \frac{1 \cdot 4 \cdot 7 \cdot 10}{3 \cdot 5 \cdot 7 \cdot 9} + \ldots + \frac{1...
  14. M

    Prime factors of a unique form in the each term a sequence?

    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...
  15. Abhishek11235

    A How to calculate the matrix of a form?

    This is screenshot from V.I Arnold's book on Classical mechanics. My question is how do we find matrix of any n-form. Detailed answer please.
  16. J

    Can These Equations Be Represented in Bloch 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...
  17. karush

    MHB Set of vectors form a vector space

    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?
  18. karush

    MHB Determine vec {{x},{y},{3x+2y}} in R^3 form a vec space

    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...
  19. Abhishek11235

    Is Every Differential 1-Form on a Line the Differential of Some Function?

    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...
  20. Mutatis

    Write ##5-3i## in the polar form ##re^\left(i\theta\right)##

    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...
  21. M

    MHB The shifting of h in vertex form

    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...
  22. R

    Alternative form of geodesic equation

    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...
  23. K

    Integral of a differential form

    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...
  24. M

    Form Factor for Scattering (like muons off of protons)

    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...
  25. B

    The squirrel jumps horizontally form the top of the 25m tall tree

    HI. I need help with my physics hw. I do not need the answers but need a general guidance on how to solve the problem. Would appreciate it alot!
  26. George Keeling

    I What is the canonical form of the metric?

    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...
  27. Krushnaraj Pandya

    Vector equation of a plane in normal form

    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...
  28. Krushnaraj Pandya

    Limit of 0^0: Evaluating x^sinx

    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...
  29. Morbidly_Green

    Expressing the density matrix in matrix form

    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?
  30. M

    MHB Exploring Basis Subsets in a 5-Element Vector Space

    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...
  31. C

    I Writing Metric in Matrix Form: Method?

    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...
  32. C

    MHB Row reduced echelon form and its meaning

    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...
  33. M

    MHB Sum of basis elements form a basis

    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...
  34. LesterTU

    Expressing Covariant Derivative in Matrix Form

    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##...
  35. C

    Bragg diffraction form an “inclined” crystal plane

    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...
  36. B

    I The integral form of Gauss' theorem

    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...
  37. V

    MHB Find Analytic Expression for Integral with Approximations

    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}...
  38. nomadreid

    I "Laws of Form" by G. Spencer-Brown (1969)

    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...
  39. J

    Maple Find Jordan Canonical Form with Maple

    Hi all! I have to show that the matrix 10x10 matrix below is nilpotent, determine its signature, and find its Jordan canonical form. [-2 , 19/2 , -17/2 , 0 , -13 , 9 , -4 , 7 , -2 , -13] [15 , -51 , 48 , -8 , 80 , -48 , 19 , -39 , 10 , 74] [-7 , 34 , -33 , 0 , -50 , 31 , -11 , 27 , -6 , -47] [1...
  40. Mr Davis 97

    Finding a closed form expression for an infinite union

    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...
  41. E

    MHB DeMoivre's Theorem express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form

    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?
  42. wolram

    B Formation of Hyperion System: How Planets Form Far from Stars

    So how did the Hyperion system form with planets so far from there star. https://www.sciencedaily.com/releases/2018/10/181015104531.htm
  43. Josu Aguirrebeitia

    A Solution form for the following differential equation

    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...
  44. E

    MHB Convert another equation x^2+y^2=4 to polar form

    x^2+y^2=4 I have so far: (r^2)cos^(theta)+(r^2)sin(theta)=4 Idk what I'm supposed to do from here
  45. E

    MHB Convert equation 8x=8y to polar form

    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
  46. Robin04

    I How to show that commutative matrices form a group?

    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?
  47. nicemaths

    Express exp(3+Pi*i) in Cartesian Form

    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...
  48. karush

    MHB 307.8.1 Suppose Y_1 and Y_2 form a basis for a 2-dimensional vector space V

    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 $V$.}$ $$Y_1=\begin{bmatrix}a\\b\end{bmatrix} \textit{ and }Y_2=\begin{bmatrix}c\\d\end{bmatrix}$$ $\textit{ then }$...
  49. Y

    MHB Show that set of points form right-angled triangle

    I was thinking of using Pythagoras here but it didn't get me far Any suggestions?
  50. M

    MHB Convert the recursive formula into the explicit form

    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)
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