Variables Definition and 1000 Threads

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.

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  1. sa1988

    Solve PDE by separation of variables

    Homework Statement Solve ∇2T(x, y) = 0 with boundary conditions T(0, y) = T(L, y) = T0 T(x, L/2) = T(x, -L/2) = T0 + T1sin(πx/L) Homework EquationsThe Attempt at a Solution Set T(x, y) = X(x)Y(y) Then ∇2T(x,y) = (∂2X/∂x2) Y + (∂2Y/∂y2) X = 0 Rearrange to find two separate ODEs: d2X/dx2 =...
  2. B

    2nd derivative change of variables

    Let's say ##f(x)=ax^2##. Then ##d^2f/dx^2=2a##. Now we can make the change of variables ##y\equiv\sqrt ax## to give ##f(y)=y^2##. Then ##d^2f/dy^2=2##. It follows that ##\frac{d^2f}{dx^2}=a\frac{d^2f}{dy^2},## but I can't replicate this with the chain rule. I would put...
  3. H

    Exploring Loop Variables: Understanding the Role of Photons in Closed Flux Loops

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  4. J

    Applied Systems of 1st order PDEs with many independant variables?

    Does anyone know of any books or online resources that do a good job discussing systems of linear 1st order PDEs with several (more than 2) independent variables? I am not a mathematician, but can handle graduate level classical physics with the associated applied math. Analytical and...
  5. I

    How to create a matrix with variables?

    Hello, I am kind of new to Matlab so the questions I will ask probably sound a bit basic. Anyways, here goes: I want to create the matrix below which has both constants and variables. How can I do this? I know how to create a normal matrix (e.g. B = [1 0 2; 3 4 5; 0 2 3]) but I don't know how...
  6. DOTDO

    Are q and q' dependent variables in Lagrangian or not?

    Hi. I have thought that the variables q and q' in L = L(q, q') are independent. (q' = dq/dt) Of course q and q' are functions of time t , but they are only dependent in terms of t . However, in the sight of general(or abstract? I mean, not specific) functional L(q, q'), q and q' are just...
  7. naima

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  8. S

    What variables affect the amplitude and equilibrium of waves and springs?

    What variables are maximum at amplitude and equilibrium? What variables are minimum at amplitude and equilibrium? I'm confused about this topic... I know KE is max at equilibrium and PE is max at amplitude.
  9. karush

    MHB *Simultaneous equation 4 variables

    Solve the following system for x and y in terms of a and b where $ab\ne - 1$ $\left\{{\frac{x+y-1}{x-y+1}=a \atop\frac{y-x+1}{x-y+1}=ab}\right.$ Answer is $x=\frac{a+1}{ab+1}\\y=\frac{a\left(b+1\right)}{\left(ab+1\right)}$ I tried for a hour and? Noticed the denominators were = but?
  10. matt_crouch

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  11. B

    Taylor expansion with multi variables

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  12. M

    Limit of function with two variables

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  13. 22990atinesh

    Solving recurrence by changing variables

    Consider the below recurrence ##T(n) = 2 T(\sqrt(n)) + \log n## substituting ##\log n = m \implies n = 2^m## ##T(2^m) = 2 T(2^{\frac{m}{2}}) + m## substituting ##S(k) = T(2^m)## I'm getting below equation ##S(k) = 2 S(\frac{k}{2}) + m## How can I change 'm' to 'k' in above equation.
  14. E

    Variables definition in Euler's introduction to analysis

    In his book, Euler gives the definition of a variable to be : "A variable quantity is an indeterminate or universal quantity, which includes within itself all completely determined values." What does he mean exactly in the last part of the sentence?
  15. morrobay

    Hidden variables and Bell's inequality

    [Mentor's note: Moved from a thread about field theories as this is just about the basic meaning of the theorem for ordinary entangled particles]] Suppose that there are photon spin outcomes that are pre existing from entanglement or from local hidden variables. For every detector angle...
  16. D

    Definable relationships between 4 variables with 2 equations

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  17. E

    Partial differentation of two variables

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  18. C

    Thermodynamic Potentials in Proper Variables

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  19. TheMathNoob

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  20. Mr Davis 97

    Multiplying or dividing an equation with variables

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  21. C

    Why are the variables different in Hibbeler's Dynamics?

    I am currently in a college dynamics class, and we are discussing energy and work. In other classes(calculus or physics), we have represented work, potential energy, and kinetic energy with the variables W, K, and U (respectively). However, in my dynamics book, these are represented using the...
  22. C

    Solving a system of equations for four variables

    Homework Statement Consider the polynomial ##p(x) = x^4 + ax^3 + bx^2 + cx + d##, where a, b, c, d ∈ ℝ. Given that 1 + i and 1 - 2i are zeroes of p(x), find the values of a, b, c and d. Homework EquationsThe Attempt at a Solution Since 1 + i and 1 - 2i are zeroes, I believe it follows that 1...
  23. TheMathNoob

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  24. M

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  25. X

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  26. L

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  27. W

    Marginal PMG of of 2 random variables with Joint PMF

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  28. T

    Absolute extrema function of 2 variables

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  29. W

    Solve PDE with separation of variables

    Homework Statement The wave equation for ψ(t, x) in 3D is ##\frac{\partial ^2 \psi}{\partial t^2}## - Δ ##\psi =0## Let ϒ(x) satisfy Δϒ = λϒ where λ<0. The x is in bold presumably to indicate it is in 3D, so represents also y and z? Show there is a solution of the form ψ(t, x) = sin(ωt)ϒ(x)...
  30. W

    How to Separate the Wave Equation into Three ODEs Using Separation of Variables?

    Homework Statement The wave equation for ψ=ψ(t,x,y) is given by ##\frac{\partial ^2 \phi}{\partial t^2} - \frac{\partial ^2 \phi}{\partial x^2} - \frac{\partial ^2 \phi}{\partial y^2}## Use separation of variables to separate the equation into 3 ODEs for T, X and Y. Use the separation...
  31. W

    Can All Differential Equations Be Separated?

    I just wanted to check something. The equation ∂2φ / ∂x2 + ∂2φ / ∂y2 = sin(xy) Was given as an example of a separable equation. I can't separate it, and I found online that to use separation of variables the equation should be linear, which this isn't? Is there a way of separating this?
  32. B

    Linear kinematics problem with limited variables.

    A rock is dropped from a cliff of height h. At the same time a rock is thrown straight up from the bottom with an initial velocity. Assuming they meet, at what time will they meet. V1 = 0 A = -9.82 Thats all you have across both rocks... is this even possible. You probably have to do some...
  33. T

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  34. C

    Constants and variables (the very basics)

    Variable; A symbol for a number we don't know yet. It is usually a letter like x or y. Example: in x + 2 = 6, x is the variable. [mathsisfun.com] Q. Why is it called the variable? This seems to me to imply that its value varies. In the above example it seems to me to simply be the unknown...
  35. A

    3D graph plot of a function of multiple variables.

    I am searching for a site that enables me to plot a function of multiple variables in 3D form. Say f(x1,x2,x3,x4) and I wish to plot this function as a 3D graph for x1 and x2 where x3 and x4 are kept constant.
  36. S

    Conceptual Problems with Random Variables and Sample Theory

    Hi I'm having a few conceptual difficulties with random variables and I was hoping someone could clear up a few things for me: 1) Firstly, what exactly do we mean when we say that two random variables X and Y are equal. I understand what identically distributed means, but my difficulty is with...
  37. 0

    Consequences of Choosing Incorrect Variables in BuckinghamPi

    This question arose as I was studying mathematical modeling in fluid mechanics. It was posted to math.stackexchange, but there a was a lack of response, probably due to the applied nature of the problem. One form of the Buckingham Pi Theorem says that for nn variables with kk dimensions, the...
  38. S

    Selecting Variables for Equation Building - Fastest Observed Velocity of Matter

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  39. S

    Separation of variables and potential

    Homework Statement A potential satisfies ##\nabla^2 Φ = 0## in the 2d slab ## -\inf < x < \inf ##, ##-b < y < b ##, with boundary conditions ## Φ(x, +b) = +V_s(x)## on the top and ##Φ(x, b) = -V_s(x)## on the bottom, where[/B] ##V_s (x)= -V_0 for -a<x<0## ##V_s (x)=+V_0 for 0<x<a## (a) what...
  40. M

    Finding two variables in projectile motion

    Homework Statement You are the chief of the Angry Birds (Google it, if you don’t know the reference...it’s not important). You can shoot red birds from a catapult, and they will fly as projectiles under the effect of gravity. Gravity points downwards and has magnitude g = 9.80 m/s2 . Your goal...
  41. yango_17

    Domain and range of a function of several variables

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  42. V

    MHB Help Understanding Probability: Random Variables & Coin Flips

    I'm going through examples in my textbook in probability and found one that I just can't follow, so I'm wondering if someone might be able to help me. Ex: If the sample space corresponds to flipping three different coins, then we could let X be the total number of heads showing, let Y be the...
  43. S

    Separation of Variables Problem

    Homework Statement A square is made up of four plates with a potential of zero on the top and bottom plates at (x,L/2) and (x,-L/2), and a potential of cos(πy/L)+cos(3πy/L). Find the potential and electric fields inside the square. The Attempt at a Solution I start with...
  44. Math Amateur

    MHB Software for Graphing Polynomials in Two Variables

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  45. A

    Change of variables jacobian limits

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  46. R

    C/C++ C++ Variable Transfer Help - Ask for Advice

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  47. Jimster41

    B Second Law and "Hidden Variables"

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  48. K

    Maple Partially Factor Lengthy Expression w/ Maple - Positive Real Numbers

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  49. B

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  50. G

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