What is Variables: Definition and 1000 Discussions

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol.
In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context.
A bound variable is a variable that was previously free, but has been bound to a specific value or set of values called domain of discourse or universe. For example, the variable x becomes a bound variable when we write:

For all x, (x + 1)2 = x2 + 2x + 1.
or

There exists x such that x2 = 2.
In either of these propositions, it does not matter logically whether x or some other letter is used. However, it could be confusing to use the same letter again elsewhere in some compound proposition. That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.
The term "dummy variable" is also sometimes used for a bound variable (more often in general mathematics than in computer science), but that use can create an ambiguity with the definition of dummy variables in regression analysis.

View More On Wikipedia.org
Recent contents Top users
  1. S

    Poisson distribution involving 2 variables

    Homework Statement A coffee shop sell tea and coffee. The number of cups of coffee sold in a minute can be assumed to be a random poisson variable with mean = 1.5 . The number of cups of tea sold can be assumed to be an independent random variable with mean = 0.5. Calculate the probablity...
  2. atyy

    Naimark extension for continuous variables

    For discrete variables, a POVM on a system can be thought of as a projective measurement on the system coupled to an apparatus. This is called the Naimark extension. Is this also true for continuous variables? http://arxiv.org/abs/1110.6815 (Theorem 4, p10)
  3. C

    MHB Separation of Variables Problem

    (2xy-3y)dx-({x}^{2}-x)dy=0 ans. xy(x-3)=C ty
  4. P

    Calculus of variations: multiple variables, functions of one variable

    Simply put, can you find the function which extremizes the integral J[f]=\iint L\left(x,y,f(x),f(y),f'(x),f'(y)\right) \,dx \,dy Where ##f## is the function to be extremized, and ##x## and ##y## are independent variables? A result seems possible by using the usual calculus of variation...
  5. M

    Separation of Variables In Electrostatics

    I am curious how legitimate a solution Separation of Variables tends to give. I've been working problems out of Griffith's book on Electromagnetism, and am often uneasy as to the way things are done. I have two specific issues. The first, is that in spherical it is often necessary to remove...
  6. T

    Change of variables for 2nd order differential

    Hi there, I'm having some difficulty in understanding how the change of variables by considering a retarded time frame can be obtained for this particular eqn I have. Say I have this original equation, \frac{\partial A}{\partial z} + \beta_1 \frac{\partial A}{\partial t}+ \frac{i...
  7. E

    Change of variables Laplace Equation

    Homework Statement Write the Laplace equation \dfrac {\partial ^{2}F} {\partial x^{2}}+\dfrac {\partial ^{2}F} {\partial y^{2}}=0 in terms of polar coordinates. Homework Equations r=\sqrt {x^{2}+y^{2}} \theta =\tan ^{-1}(\frac{y}{x}) \dfrac {\partial r} {\partial x}=\cos \theta \dfrac...
  8. R

    Multiplication of conditional probability with several variables

    Dear All, I am a starter to machine learning and i am currently confused about the following problem: what is the result of P(X|Y)P(Y|Z)? In my book, it is written to be P(X|Z). But I don't think it is correct since P(X|Z)= P(X|Y,Z)P(Y|Z) But clearly P(X|Y)=/= P(X|Y,Z) Assuming...
  9. P

    Solving a trigonometry equation simultaneously in two variables

    Homework Statement Solve the equations sin(x+2y)=1/2 , cos(2x-y)=1/ sqrt(2) Homework Equations The Attempt at a Solution I tried getting a generic solution for both the first and second equation. How do I further proceed? By simultaneously solving? I so, how? 2x-y=2m*pi + or...
  10. M

    How would I correlate many variables to a few coefficients?

    I have around 550 asymmetrical sigmoid curves fitted to a function with 4 varying coefficients. Each of these curves represent strength as a function of time and temperature for a different compound. Each compound is made up of varying substances at varying concentrations. Overall, I have 550...
  11. D

    How to Analyze Correlation Between Variables

    Homework Statement I'm doing a research project in which there's a survey given to respondents from 3 different groups of education(high school vs university etc) related to STD contraction rates. On the survey there's a question asking how many STD's they have contracted over their lifetime...
  12. 1

    Cylinder with heat generation, Separation of variables

    I'm having some difficulty setting up a problem. I'm trying to model the temperature of a thermistor connected to a constant current source. The thermistor's resistance varies with temperature, so with a fixed current, I would expect to see the thermistor's temperature to oscillate with time...
  13. M

    Proving a dependence between variables

    Homework Statement I want to find the relationship between ##\psi## and ##\gamma## from this equation $$\Lambda=\sqrt{\gamma ^2+4 \pi ^2 \psi ^2} \sum _{j=0}^{\infty } \frac{\left(\frac{(2 j-1)\text{!} (2 \pi \psi )^j}{(2 j)\text{!} \left(\gamma ^2+4 \pi ^2 \psi...
  14. estro

    How Does the Symmetry of Sine Influence the Distribution of Y = sin(X)?

    Suppose X ~ U[ 0, pi ] What is the distribution of Y=sinX. I have a solution in my notes however I don,t understand the following the second transition: F_Y(y) = P(Y \leq y) = P(X \leq \arcsin(y)) + P(X \geq \pi - \arcsin(y)) = ... Where the P(X \geq \pi - \arcsin(y)) comes from?
  15. A

    MHB Equation of tangents and separation of variables question

    Hey guys, Can anyone help me out with these questions? The first one has a positive initial value. Separation of variables and integrating gave me: |y+3| = k√[(t^2) + 1)] Ultimately, I got k= √5 and thus y=√5√[(t^2) + 1)] - 3. Also, for the second one, I used a similar process, found...
  16. C

    Positive probabilities for neg sums of uniformly distributed variables

    I've been thinking about the Central Limit Theorem and by my understanding it states that the sum of randomly distributed variables follows approximately a normal distribution. My question is if you have, say, 100 uniformly distributed variables that range from 0 to 10, their sum has to be...
  17. R

    MHB What Are the Variable Restrictions in the Equation 5/x = (10/3x) + 4?

    Write the restrictions on the variables for the following equation. Keeping in mind the restrictions, solve the equation 5/x=(10/3x)+4 Here is what I did, but I'm not sure if it's right. (3x)(5/x)=(3x)[(10/3x)+4] 15=(3x)(10/3x)+(3x)4 15=10+12x 5=12x x=5/12 But I don't know what to do...
  18. L

    Hidden Variables and Quantum Mechanics

    I'm reading up on interpretations of quantum theory, and I just came across Bell's theorem, which is confusing me. My main concern is this: Why would quantum mechanics predict something different than hidden variables? I hope that question is coherent enough. I'm not sure if I'm using the...
  19. K

    MHB Inequality involving fractions and several variables

    What are some simplified conditions for which: $$W\bigg(A-\frac{X}{W}\bigg)^3\bigg[X-AW-\frac{AY}{N}(B+D)-\frac{AZ}{N}(C+D+E+F+G)\bigg]+\frac{X}{N}\bigg[Y(A+H)(B+D)+AZ(C+D+E+F+G)\bigg]<0$$ **WHERE:** All of the letters are positive parameters (not constants) and: $1.$ $$A,B,C,D,E,F,G,H < N...
  20. B

    Change of variables in double integrals

    I know the formula for a change of variables in a double integral using Jacobians. $$ \iint_{S}\,dx\,dy = \iint_{S'}\left\lvert J(u,v) \right\rvert\,du\,dv $$ where ## S' ## is the preimage of ## S ## under the mapping $$ x = f(u,v),~ y = g(u,v) $$ and ## J(u,v) ## is the Jacobian of the mapping...
  21. B

    Function is a change of variables?

    Hi there! The question is: if I have to prove that a function is a change of variable it is sufficient to prove that the function is a diffeomorphism? i.e. prove that the function is bijective, differentiable, and its inverse is differentiable? Thanks!
  22. T

    Power Expansion (Complex variables)

    Homework Statement Use the power series for e^z and the def. of sin(z) to check that sum ((-1)^k z^(2 k+1))/((2 k+1)!) Homework Equations The Attempt at a Solution I apologize, but I am not particularly good with latex. Therefore, I attached a picture of my solution thus far...
  23. PsychonautQQ

    Polynomials in n variables subspaces and subrepresentations

    Homework Statement Trying to make sense of my notes... "A polynomial in n variables on an n-dimensional F-vector space V is a formal sum of the form: p(x)= ∑(C_i)x^β" so basically can somebody help me understand how polynomials represent vector spaces? Whatever degree the polynomial is...
  24. W

    Newton-Raphson Method for Non-linear System of 3 variables in Matlab

    I am trying to solve 3 non-linear system of 3 variables using the Newton-raphson method in matlab. Here are the 3 non-linear equations: \begin{equation} c[\alpha I+ k_f+k_d+k_ns+k_p(1-q)]-I \alpha =0 \end{equation} \begin{equation} s[\lambda_b c P_C +\lambda_r (1-q)]- \lambda_b c P_C =0...
  25. R

    MATLAB Newton-Raphson Method for Non-linear System of 3 variables in Matlab

    I am trying to solve 3 non-linear system of 3 variables using the Newton-raphson method in matlab. Here are the three equations: \begin{equation} c[\alpha I+ k_f+k_d+k_ns+k_p(1-q)]-I \alpha =0 \end{equation} \begin{equation} s[\lambda_b c P_C +\lambda_r (1-q)]- \lambda_b c P_C =0 \end{equation}...
  26. Y

    MHB Solving a Tricky Chain Rule Question with Confusing Variables

    Hello, I have a tricky chain rule question, I think understanding it is more difficult than solving. For the function z=f(x,y) it is given that: f_{y}(0,-3)=-2 and \[f_{x}(0,-3)=3\] so for the function \[g(x,y)=f(2\cdot ln(x+y),x^{4}-3y^{2})\] choose the correct answer: (1)...
  27. D

    MHB Calculate E(g(X)) for Random Variable X with E(X)=6.2, Var(X)=0.8

    This problem: A random variable X has expected value E(X) = 6.2 and variance Var(X) = 0.8. Calculate the expected value of g(X) where g(x) = 7x + 2. Do I just plug in numbers here? I've never seen this kind of problem before.
  28. G

    Hidden variables of polarisers

    This post is my reply to a blog comment by billschnieder, since I do not think it is wise to start a discussion in the blog's comments directly, nor in private messages, because of both the reduced visibility of the discussion and the increased spam-like characteristics of such communication for...
  29. J

    Interpolation with 2 variables

    If given three points ##P_0 = (x_0, y_0)##, ##P_1 = (x_1, y_1)## and ##P_2 = (x_2, y_2)##, the polynomial function ##f(x)## that intersect those points is ##f(x) = a_2 x^2 + a_1 x^1 + a_0 x^0##. where: ## \begin{bmatrix} a_0\\ a_1\\ a_2\\ \end{bmatrix} = \begin{bmatrix} x_0^0 & x_1^0 & x_2^0...
  30. R

    MHB Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s a

    The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 \end{equation} \begin{equation} q[2.73 + (5.98*10^{10})c]-(5.98 *10^{10})c =0 \end{equation} Using the...
  31. C

    Finding the Second Partial Derivative of a Multivariable Function

    Homework Statement Show z(x,y) = cos(xy) is a solution of (∂z/∂x)y + (∂z/dy)x = (x+y) ( (∂2z/∂x∂y) + xyz) (question also attached if it makes it clearer) The Attempt at a Solution ∂z= (∂z/∂x)ydx + (∂z/dy)xdy ∂z/∂x = -ysin(xy) ∂z/∂y = -xsin(xy) what does it mean show it...
  32. H

    How do you reduce a matrix with unknown components?

    Hi, I've been running into a problem lately where I have a system of equations that needs to be solved or I need to do some other sort of matrix algebra, but the components of the matrix that I am trying to perform row operations on have unknowns in them. Specifically, I was working with a...
  33. V

    Double integral change of variables

    Homework Statement Use the change of variables ##u=x+y## and ##y=uv## to solve: \int_0^1\int_0^{1-x}e^{\frac{y}{x+y}}dydx Homework Equations The Attempt at a Solution So I got as far as: \int\int{}ue^vdvdu. But I just can't find the region of integration in terms of ##u## and ##v##.
  34. M

    Simple problems regarding sum of IID random variables

    Hi! I'm taking my first course in statistics and am hoping to get some intuition for this set of problems... Suppose I have a bowl of marbles that each weighs m_{marble}=0.01 kg. For each marble I swallow, there is a chance p=0.53 that it adds m_{marble} to my weight, and chance 1-p that...
  35. O

    Roman vs. Greek letters for variables

    I am writing a paper and I was using 'o' for a variable. Then I decided that it looked too much like '0', and thought I might use capital omega. How do I know when I should and should not use Greek letters? Thanks
  36. S

    Help with linear transformation problem with variables

    Let L: R3 -> R3 be L(x)= \begin{pmatrix} x1+x2\\ x1-x2\\ 3x1+2x2 \end{pmatrix} find a matrix A such that L(x)=Ax for all x in R2 From what I understand I need to find the transition matrix from the elementary to L(x). However it is'nt a square matrix and it has variables instead of numbers...
  37. S

    Change of Variables with Jacobians

    Homework Statement Suggest a substitution/transformation that will simplify the following integral and find their jacobians: \int\int_{R}x\sin(6x+7y)-3y\sin(6x+7y)dA Homework Equations [ tex ] \int\int_{R}x\sin(6x+7y)-3y\sin(6x+7y)dA [ / tex ] The Attempt at a Solution This topic is...
  38. J

    Can a vector field be represented as a surface in a vector space?

    For example the surface (x,y,x²+y²), can for example surfaces be considered as one abstract 'vector' in some abstract 'vector'-space? The ' ' because surfaces might not be a vector space. For surfaces we can exceptionally define normal vectors at every point.
  39. Y

    Change of variables to evaluate the integral

    Homework Statement these are on the picture Homework Equations transformation, jacobian The Attempt at a Solution I don't know how to enter the equation, so i uploaded the picture.. is it alright? I think I solved it somehow, but not quite sure if it is right... please tell me if there's a...
  40. J

    Exploring Different Forms of Polynomials in Two Variables

    If a polynomial of 1 variable, for example: P(x) = ax²+bx+c, can be written as P(x) = a(x-x1)(x-x2), so a polynomial of 2 variables like: Q(x,y) = ax²+bxy+cy²+dx+ey+f can be written of another form?
  41. M

    Second order PDE (w.r.t 2 variables)

    Homework Statement find the solution to: \frac{\partial^{2}u}{\partial x \partial y} = 0 \frac{\partial^{2}u}{\partial x^{2}} = 0 \frac{\partial^{2}u}{\partial y^{2}} = 0 Homework Equations theorem of integration The Attempt at a Solution now from a previous question I...
  42. B

    Partial differentiation problem, multiple variables (chain rule?)

    Homework Statement if z = x2 + 2y2 , x = r cos θ , y = r sin θ , find the partial derivative \left(\frac{\partial z}{\partial \theta}\right)_{x} Homework Equations z = x2 + 2y2 x = r cos θ y = r sin θ The Attempt at a Solution The textbook says that the equation should be...
  43. J

    Separation of variables for solutions of partial differential equation

    Why is it assumed that the method of separation of variables works when the boundary conditions of some boundary valued problem are homogeneous? What is the reasoning behind it?
  44. P

    What are the variables used in laser equations?

    Homework Statement hello i need a pdf or a homepage where i can get all symbols (variables ) for Laser questions for example λ=Wavelength v=frequency ω=Beam waist Q= quality factor etc. i have a problem understanding some questions because i don't know which symbol they...
  45. B

    Noether's Theorem For Functionals of Several Variables

    My question is on using a form of the single variable Noether's theorem to remember the multiple variable version. Noether's theorem, for functionals of a single independent variable, can be translated into saying that, because \mathcal{L} is invariant, we have \mathcal{L}(x,y_i,y_i')dx =...
  46. K

    Density function of product of random variables

    suppose you have two random variables X and Y which are independent, we want to form a new random variable Z=XY, if f(x) and f(y) are density functions of X and Y respectively what is the density function of Z? I tried taking logs and applying convolution, but it did not really work
  47. S

    Diffusion equation by seperation of variables

    Homework Statement A uniform rod of length l has an initial (at time t = 0) temperature distribution given by u(x, 0) = sin(\frac{πx}{l}), 0 \leq x \leq l. The temperature u(x, t) satisfies the classical one-dimensional diffusion equation, ut = kuxx The ends of the rod are...
  48. J

    Independent variables of the Lagrangian

    Why are y and y' treated as independent variables, while they are not? Another slightly related question: if ' = d/dt then df'/dg' = df/dg because f' = df/dg g', but if we differentiate f' to g' we implicitly assume that df/dg is independent of g', is it?
  49. M

    MHB How to solve this boundary value problem-Method of separation of variables

    Hey! :o I have a question.. (Wasntme) When we have the following boundary value problem: $$u_{xx}+u_{yy}=0, 0<x<a, 0<y<b (1)$$ $$u_x(0,y)=u_x(a,y)=0, 0<y<b$$ $$u(x,0)=x, u_y(x,b)=0, 0<x<a$$using the method of separation of variables, the solution would be of the form $u(x,y)=X(x) \cdot Y(y)$...
  50. MarkFL

    MHB David's Math: Solving a Non-Linear System in 2 Variables

    Here is the question: I have posted a link there so the OP can view my work.
Back
Top