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. gracy

    Form taken by potential energy

    I have come across the following paragraph from my book "The gravitational force is always attractive .Whereas the electric force is attractive or repulsive acording to whether q0 is negative or positive.Each of the above forces is conservative,so a potential energy is associated with each of...
  2. Q

    Gauss' law in differential form

    I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that ## \nabla\cdot E=\frac{1}{\epsilon 0}\rho##, but when I calculate it directly I get that ## \nabla\cdot E=0## (at least for ## r\neq0##). 2) Now ##...
  3. D

    Deriving Riemann Tensor Comp. in General Frame

    How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...
  4. M

    How to convert Euler Equations to Lagrangian Form?

    I am not entirely sure how to convert the conservation of mass and momentum equations into the Lagrangian form using the mass coordinate h. The one dimensional Euler equations given by, \frac{\partial \rho}{\partial t} + u\frac{\partial \rho}{\partial x} + \rho\frac{\partial u}{\partial x} = 0...
  5. P

    Matrix Representation of a Uniform Sphere Centered at the Origin

    What is the basic matrix form for a uniform (unit) sphere centered at the origin? Given a vector that specifies the radii (1,1,1) == (r1,r2,r3), I would like the matrix that implies no rotation (is it [[1,0,0],[0,1,0],[0,0,1]]?) and covers the rest of the necessary parameters. I am testing...
  6. UMath1

    Why do polyatomic ions form and how are they stabilized?

    I understand that monoatomic ions such as Sodium and Magnesium form to fill there outer shells, but why do polyatomics form? I know that CO2 is a pretty stable compound, so why does carbonate even form? Likewise what leads to the formation of ammonium?
  7. S

    Parametric vector form of cartesian equation

    How can I find the parametric vector form of a cartesian equation under a specific condition? Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1...
  8. Aristotle

    How to find decimal of a number in One's Complement form?

    Homework Statement Assume the number https://www.physicsforums.com/tel:1111111 Is in one's complement form, what is its decimal values? Homework Equations 2^7 + 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0 1 in first sign bit indicates negative 0 in first sign bit indicates positive The Attempt...
  9. A

    Putting an equation into y = mx+b form

    Homework Statement I am doing an experiment where I have to test the validity of the equation: (m1-m2)g=(m1+m2+I/R^2)a. The lab instructions say to linearize this equation. What would be the "m" and "b" values? Homework Equations y=mx+b (m1-m2)g=(m1+m2+I/R^2)a The Attempt at a Solution I...
  10. B

    How to form the transformation matrix for this

    We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis. The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or...
  11. B

    Form factored of the polynomial discriminant

    I wrote x² - (a + b)x + (ab) in the wolfram and polynomial discriminant was: a² - 2ab + b². Factoring: (a-b)² --- So, I wrote x³ - (a+b+c) x² + (bc+ca+ab) x - (abc) and the polynomial discrimant given was: Factoring: (b-c)² (c-a)² (a-b)² --- Now, I wrote x² - 2Ax + B² and the polynomial...
  12. Bill_Nye_Fan

    Simplify x^(n-1)⋅³√(y^2/2x^5) into Rationalised Surd Form

    Homework Statement Q7.[/B] a) Express x^(n-1)⋅³√(y^2/2x^5) in its simplest, rationalised surd form. b) Given that the solution to part a) is 5, and that y can be expressed as 1/x^(6n+5)/4), determine the value of x. Again, express your answer in rationalised surd form. *Note, this is a...
  13. Priyadarshini

    How Do You Tell When A Compound Will Form A Coordinate Bond?

    How can you tell when a compound will form a covalent bond or a coordinate bond? I know that a coordinate bond is a special type of covalent bond and if during covalent bonding, if the elements taking part do not obtain a noble gas configuration, they for coordinate bonds. But take for example...
  14. M

    MHB Why does $Y_n$ have that specific form?

    Hey! :o Let $R$ be any integral domain of characteristic zero. We consider the Pell equation $$X^2-(T^2-1)Y^2=1\tag 1$$ over $R[T]$. Let $U$ be an element in the algebraic closure of $R[T]$ satisfying $$U^2=T^2-1\tag 2$$ Define two sequences $X_n, Y_n, n=0, 1, 2, \dots $, of polynomials in...
  15. S

    Chern-Simons form of the Chern character

    So I'm working my way through Nakahara Geometry Topology and Physics. In exercise 11.4 of section 11.5.2 I'm tasked to show...
  16. mattyk

    System of Linear Equations to Reduced Echelon Form

    I have x1 + 3x2 - x3 = b1 x1 - x2 + 3x3 = b1 -2x1 - 5x2 - x3 = b1 So using an augmented matrix I get this [1 3 -1 | 1] [1 -1 3 | 1] [-2 -5 1 | 1] [1 3 -1 | 1] [0 -4 4 | 0] R2 - R1 = R2 [0 1 -1 | 3] R3 + 2R1 = R3 [1 3 -1 | 1] [0 1 -1 | 3] Swap R2 with R3 [0 -4 4 | 0] [1 0 2 | -8] R1 - 3R2 =...
  17. Corey Bacon

    Net force on particle 1 due to particle 2 in vector form?

    Homework Statement Hi all, I have this quiz on MasteringPhysics, but I can't seem to get the right answer.[/B] Consider two positively charged particles, one of charge q0 (particle 0) fixed at the origin, and another of charge q1 (particle 1) fixed on the y-axis at (0,d1,0). What is the net...
  18. emma69

    Do grasses prefer their nitrogen in the nitrate form?

    "Most vegetables, annuals and grasses prefer their nitrogen in the nitrate form and as such do better in alkaline inclined soils dominated by bacteria. Most trees, shrubs and perennials prefer their nitrogen in the ammonium form and as such do better in acid inclined soils dominated by fungi."
  19. T

    How Do You Transform Trigonometric Expressions and Find Their Extremes?

    Homework Statement Express $$4sin\theta-3cos\theta$$ in the form $$rsin(\theta-\alpha)$$ Hence find the maximum and minimum values of $$\frac{7}{4sin\theta-3cos\theta+2}$$ State the greatest and least values. Homework EquationsThe Attempt at a Solution Okay so putting it in the...
  20. M

    Differential Form - Notation Help

    Hi there, I was reading up on Holonomic constraints and came across this equation on the Wikipedia page: The page says it is a differential form. Can anyone explain the notation for me or provide a link or two to documents or pages which explain this notation? Thank you very much, Geoff
  21. TrickyDicky

    Implications of the tensorial form of the EFE

    The EFE are tensor equations in 4-dimensional spacetime and by virtue of their tensorial form indepedence from the choice of coordinate system is guaranteed, and the same goes for the metric tensor solutions. When looking for assumptions that help simplify the process of solving the EFE to find...
  22. S

    Understanding // in the Hint for Showing Numbers of Form ±m√2/n Are Dense

    Hello, Please, someone, explain what the // in the hint below stands for: "Show that the numbers of the form ±m√2/n for m, n ∈ N are dense." Hint: "To find a number in (x, y), find a rational in (x//√2, y//√2). Conclude from this that the set of all (irrational) numbers of the form ±m√2/n is...
  23. avito009

    Is Proton Mass Really Reduced in Helium Nuclei?

    I am no Einstein but I would like to expand my knowledge and share it. When two deuterium atoms fuse together they become a helium nuclei. Now deuterium nuclei contains 1 proton and 1 neutron. When the deuterium nuclei fuses to form a helium nuclei. Helium nuclei contains 2 protons, and 2...
  24. P

    Matrix form of Density Operator

    Hi All, I have spent hours trying to understand the matrix form of Density Operator. But, I fail. Please see page 2 of the attached file. (from the book "Quantum Mechanics - The Theoretical Minimum" page 199). Most appreciated if someone could enlighten me this. Many thanks in advance. Peter Yu
  25. Cosmophile

    Concerning Vectors in Scalar Form

    Hey, all. I have a question concerning the treatment and use of vectors when solving problems (or in general, really). I know that vectors have both magnitude and direction, while scalars only have magnitude. However, in solving problems and looking at how others have solved them, I've noticed...
  26. P

    Inverse hyperbolic functions (logarithmic form)

    To express the ##\cosh^{-1}## function as a logarithm, we start by defining the variables ##x## and ##y## as follows: $$y = \cosh^{-1}{x}$$ $$x = \cosh{y}$$ Where ##y ∈ [0, \infty)## and ##x ∈ [1, \infty)##. Using the definition of the hyperbolic cosine function, rearranging, and multiplying...
  27. S

    Integral form of Maxwell equations.

    I have been studying the Maxwell equations recently (namely the integral forms of them). Of course I had to study line integrals before that. Well, I went to a hyperphysics page to look up the equations: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/maxeq.html I noticed that the...
  28. evinda

    MHB Why is the general solution of this form?

    Hello! (Wave)I found the following in my lecture notes:$$u_t=u_{xx}, x \in \mathbb{R}, t>0 \\ u(x,0)=f(x)$$ $$u(x,t)=X(x)T(t)$$ $$\Rightarrow \frac{T'(t)}{T(t)}=\frac{X''(x)}{X(x)}=-\lambda \in \mathbb{R}$$ $$X''(x)+\lambda X(x)=0, x \in \mathbb{R}$$ $$X \text{ bounded }$$ The characteristic...
  29. abm77

    Four charges equal in magnitude form a square

    Homework Statement Four charges equal in magnitude of 20.0 microC are placed on the four corners of a square with side length 0.180m. Determine the electric field at the centre of the square. (-q) ---------- (+q) l l l l l...
  30. diracdelta

    Quadratic form and diagonalization

    Homework Statement Find diagonal shape of next quadratic form ( using eigenvalues and eigenvectors) Q(x,y)= 5x2 + 2y2 + 4xy. What is curve { (x,y)∈ ℝ| Q(x,y)= λ1λ2, where λ1 and λ2 are eigenvalues of simetric matrix joined to quadratic form Q. Draw given curve in plane. The Attempt at a...
  31. S

    Covariant form of Newton's law of gravity

    Hii! Newton's law of gravity is ∇.(∇Φ) = 4πGρ. A book on GR gives a suggestion to make it Lorentz covariant by using de' Alembertian operator on 'Φ' in the LHS of above equation instead of Laplacian. Then it explains that this won't work because we have to include in 'ρ' all the energy...
  32. S

    Proof about system of linear equations in echelon form

    Homework Statement Problem: Consider a system of linear equations in echelon form with r equations and n unknowns. Prove the following.: (i) If r = n, then the system has a unique solution. (ii) If r < n, then we can arbitrarily assign values to the n - r free variables and solve uniquely...
  33. T

    Minimum force required to form sphere

    Homework Statement Homework EquationsThe Attempt at a Solution Doing a vertical force balance 2Fcosθ=mg ,where m is the mass of water . Not sure how to proceed . What role does the pin hole at the top play ? I would be grateful if somebody could help me with the problem.
  34. C

    MATLAB Transforming Complex Exponential to Discrete Vector Form

    Hi, I want to transform a complex exponential with quadratic phase to discrete form, in other words to a vector form. can anyone help me with that? Thanks
  35. Syed Alam

    Enthelpy rise hot channel factor and Radial form factor

    I am currently designing a whole-core for small PWR. I am calculating core life (years) vs. FdH and RFF. What is the difference between "Enthalpy rise hot channel factor (FdH)" and "Radial form factor (RFF)"? I have calculated "CHANNEL FDH" which is same as "RFF". Do I have to try "CHANNEL...
  36. Prashan Shan

    What are the different forms of Virtual Photons?

    1).Virtual Photons forms in pairs like particles and anti particles? or 2).by borrowing energy from future? or 3) in both ways?
  37. S

    Integral form of Poisson's equation

    Is it true that ##V(\textbf{r}) = \frac{1}{4 \pi \epsilon_0} \int \frac{1}{|\textbf{r}-\textbf{r}^{'}|}\ \rho(\textbf{r}^{'})\ d \tau^{'}## is the integral form of Poission equation ##\nabla^{2} V = - \frac{1}{\epsilon_0} \rho##? Can you trace the steps that lead from one to the other, or at...
  38. kostoglotov

    Laplace in Cyl. form: one step I'm not sure about

    I have completed the exercise, but I did something weird in one step to make it work, and I'd like to know more about what I did...or if what I did was at all valid. 1. Homework Statement Show that Laplace's equation \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial...
  39. A

    Operators change form for density matrix equations?

    Imagine applying an operator to a wave-function: \psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||} Where ## \psi _t(x_1, x_2, ..., x_n) ## is initial system state vector, denominator is normalization factor, and Ln(x) is a...
  40. U

    Bernoulli vs conservation of momentum (Reynolds transport theorem form)

    So I've found several instances in which Bernoulli and Conservation of momentum (in Reynolds transport theorem form) give different answers for the study of an inviscid fluid. Let's consider a simple situation as described in my diagram attached. Basically a tap/faucet is releasing fluid, which...
  41. V

    MHB Differential form surface integral

    Question: Evaluate the surface integral $$J = 2xzdy \land dz+2yzdz \land dx-{z}^{2}dx \land dy$$ where S \subset {\Bbb{R}}^{3} is the rectangle parametrised by: $$x(u,v) = 1-u,\ y(u,v) = u,\ z(u,v) = v,\ \ 0\le u, v \le 1$$ so far I have: \begin{array}{}x = u\cos v, &dx = \cos v\, du -...
  42. A

    Why do collapse functions change form in master equations?

    In the GRW spontaneous collapse model (for example) the wave-function evolves by linear Schrödinger equation, except, at random times, wave-function experiences a jump of the form: \psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||}...
  43. C

    Solving Row Echelon Form: Practicing Tips for Students

    For some reason I just can't seem to wrap my head around the idea of reducing a Matrix to row echelon form. I'm familiar with the steps that the textbooks and tutorials use and how it's done but when I try practicing on my own I feel lost. e.g. all I end up with are just a bunch of random...
  44. N

    Integral Form of Gauss' Law at Center of Finite Wire

    At the exact center of a finite wire (i.e. a distance, say $L/2$ from each end), why can I not apply Gauss's Law in integral form to find an EXACT solution for the electric field? At the center of the wire, $E$ is entirely radial, so it seems like I should be able to draw an infinitesimally...
  45. L

    Question about some general potential form

    In paper Phys. Rev. B 29, 3153 – Published 15 March 1984 general potential form is introduced and from that form one can obtain different class of period potential V(u,r)=A(r)\frac{1+e\cos (2\pi u)}{[1+r^2+2r\cos (2\pi u)]^p} ##-1<r<1## , where ##r## is real number, ##m,p## are integers...
  46. ShayanJ

    Forming Spherically Symmetric Metric: Math Analysis & Omitted Steps

    In most GR textbooks, the general form of a spherically symmetric metric is obtained by inspection which is acceptable. But in the textbook I'm reading, the author does that with a mathematical analysis just to illustrate the method. But I can't follow his calculations. In fact he omits much of...
  47. I

    Homogeneity of space and the form of the Lagrangian

    I was reading that the homogeneity of space can lead to the conclusion that the lagrangian of a free particle is not explicitly dependent on its position. At the moment, this does not come very intuitively to me. By homogeneity, I understand that if you displace the initial position of a...
  48. TheLibraSign

    MHB What is the slope and intercept in slope-intercept form?

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  49. S

    Projection stereographic and second fundamental form

    Let r:R2 →R3 be given by the formula Compute the second fundamental form with respect to this basis (Hint: There’s a shortcut to computing the unit normal n). I can't find thi shortcut, does anyone help me? I'm solving it with normal vector and first and second derivate, but I obtained...
  50. binbagsss

    System of first order equations, matrix form, quick question

    Question: ##h_{t}+vh_{x}+v_{x}h=0## ##v_{t}+gh_{x}+vv_{x}=0## Write it in the form ##P_{t}+Q_{x}=0##, where ##P=(h,hv)^{T}##, where ##g## is a constant ##>0##, and ##v## and ##h## are functions of ##x## and ##t##. Attempt: I have ##Q=(vh,?)^{T}##, the first equation looks easy enough, but...
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