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. D

    MHB Change of variables in a function

    Given $$ u(\xi,\eta) = F(\xi) + G(\eta) $$ and $$ \xi = x+ct\qquad\qquad \eta = x -ct. $$ How do we get to $u(x,t) = F(x+ct) + G(x-ct)$? I see that we can make the sub $$ u(x+ct,x-ct) = F(x+ct) + G(x-ct) $$ but how do I then get simply $u(x,t)$?
  2. N

    Peskin & Schroeder p. 285, change of variables integration measure

    I had a question about about the integration measure for the path integral after a unitary change of variables. First they consider a 4D spacetime lattice with volume L^4. The measure is \mathcal{D}\phi = \prod_i d\phi(x_i) They expand the field variables in a Fourier series...
  3. G

    Conditional expectation of a product of two independent random variables

    Suppose that α and β are independently distributed random variables, with means; μ_α, μ_b and variances; δ_α^2, δ_β^2, respectively. Further, let c=αβ+e, where e is independently distributed from α and β with mean 0 and variance δ_e^2. Does it hold that E(αβ | c) = E(α|c)...
  4. A

    Random Variables and Transformations

    In the last question in this link: http://pages.uoregon.edu/csinclai/teaching/Fall2009/files/hw8.pdf 1) I did not understand how they got the region for y1, y2, and y3... 2) How would the solution be different (or not possible) if X1, X2, and X3 were not iid? Thanks in advance
  5. S

    Electrostatic Separation of Variables in a Square Pipe

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  6. iVenky

    What is the pdf of multiplication of two random variables?

    We have two independent random variables X and Y whose pdfs are given as f(x) and f(y). Now when you multiply X and Y you get a random variable say Z. Now what is the resulting pdf f(z)? I mean how is that related to the pdf of f(x) and f(y)? From what I read it looks like f(z)=f(x) *...
  7. 5

    Third degree Taylor polynomial in two variables

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  8. A

    Implicit differentiation help three variables

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  9. Vorde

    Epsilon-Delta Limit in Two Variables

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  10. D

    Comp Sci Fortran: How to work out if 3 variables make a right angled triangle

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  11. S

    Matlab: creating a table of equations using multiple variables

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  12. ElijahRockers

    Engineering Inductor Circuit, separation of variables

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  13. D

    Trapezoid - Calculus of Several Variables

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  14. A

    Max and Min of a function of four variables

    Homework Statement Find the maximum and minimum values of the function f(x,y,z,t)=x+y+z+t subject to the constraint x^2+y^2+z^2+t^2=400. Homework Equations I think the Lagrange multiplers can be used ∇f=λ∇g The Attempt at a Solution So I found ∇f=<1,1,1,1> and ∇g=<2x,2y,2z,2t> and...
  15. E

    Explain p.d.f. of the sum of random variables

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  16. B

    Derivative with constant variables

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  17. I

    Placing random variables in order

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  18. dbmorpher

    Are Special Relativity variables transformable

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  19. A

    MHB Solving a Quadratic Equation with Two Variables

    Solve for $x$: $$4x^2 +4xy - 6y - 9 = 9 $$
  20. C

    How Do You Apply Chain Rule for Functions in Polar Coordinates?

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  21. M

    How to identify the state variables of a circuit?.

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  22. L

    Finding the limit for this function of two variables

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  23. L

    Declaring the domain of a function of two variables.

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  24. N

    Finding Volume triple integral by change of variables

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  25. N

    Double integral, change of variables or no

    Homework Statement ∫∫Se2x+3ydydx where S is the region |2x|+|3y|≤ 1 Homework Equations The Attempt at a Solution So I've done this two ways and gotten two different answers and I'm not sure which is right. I used change of variables where where u=3y+2x and v=3y-2x and I got an...
  26. Jalo

    Limit of a function of two variables

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  27. J

    How do you find the uncertainty in a function of two variables?

    I'm trying to make up an automatic spreadsheet for my Physics students so they can enter the data and see everything calculated for them, but I ran into a snag. I have two random variables, h and t with uncertainties Δh, and Δt, respectively. The uncertainty in h is simply estimated, based...
  28. tomwilliam2

    Discharging a capacitor, how variables change over time

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  29. B

    Analyzing 2D Fin Temperature with Separation of Variables

    Homework Statement We have a 2D fin that has length L (x-axis), and thickness t, (y-axis). The left side has a fixed temperature, the right side is insulated, and the top and bottom surfaces are subject to convection. Find an analytical solution for the temperature at steady state. Homework...
  30. D

    Solving limit problems for two variables

    I just encountered a limit-problem [1] which is supposed to lack a limit: x*y*e^(-(x+y)^2) as x^2+y^2 approaches infinity and e is the base of the natural logarithm. My approach was to express x and y in terms of polar coordinates, where x = r*cos(t) and y= r*sin(t). I got the following...
  31. P

    Why Might Plotting log c Against log d Not Be the Best Test for Proportionality?

    Hi, The question is the following: Another student proposes that c is proportional to d0.5 State a suitable graph that can be plotted to test this proposal. I suggested plotting log c = 0.5 log d. One of the answers the markscheme accepts is log c = log d Can anyone tell me why my...
  32. X

    Simple Gradient Question (funtion of two variables)

    Homework Statement Find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. g(x,y) = x2/2 - y2/2; (√2, 1)Homework Equations ∇f = (∂f/∂x)i + (∂f/∂y)jThe Attempt at a Solution ∇g = <x, -y> ∇g(√2, 1) = <√2, -1>...
  33. D

    MHB Verify Change of Variables for System of Equations w/ Constant Coefficients

    For the system of equations $$ \mathbf{x}' = \begin{pmatrix} a\cos 2t & a\sin 2t\\ a\sin 2t & -a\cos 2t \end{pmatrix}\mathbf{x}, $$ verify that under the change of variables: $$ \mathbf{x} = \begin{pmatrix} \cos 2t & \sin 2t\\ \sin 2t & -\cos 2t \end{pmatrix}\mathbf{u}, $$ the equations for $u$...
  34. M

    Separation of variables, laplace equation

    Homework Statement Two coaxial pipes(radius a) of the same diameter with a small gap between them are maintained at a potential difference V. By separation of variables and fitting coefficient, guess the potential within the pipe, near the gap. Homework Equations The Attempt...
  35. D

    Implicit functions in R2 vs. functions of two variables in R3

    Hello, I often encounter some confusion. For example, we have the equation for a circle of radius one given by: x2+y2=1 Now we can't express this as a function in R2 because it's a relation. however, we can still find derivatives like dy/dx or dx/dy using implicit...
  36. J

    Comp Sci Creating ifstream variables at run time in C++?

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  37. K

    Finding the maximum height given multiple variables

    Homework Statement A 4.27-kg object is thrown vertically upward from the surface of Mercury, where the acceleration due to gravity is g1 = 3.70 m/s2. The initial velocity is v1, and the object reaches a maximum height of y1. What is the maximum height, y2, if the object is thrown with a...
  38. B

    Problems with propagation of error for multiple variables

    please help by telling me whether my approach to solve the problems are right or wrong. please refer to the ATTACHMENT for the questions and my approaches... your help is highly appreciated!
  39. Q

    Find (∂V/∂T) of a Van der Waals Gas using variables?

    Find (∂V/∂T) of a Van der Waals Gas using variables? Evaluating a Derivative of the van der Waals Equation using the Cyclic Rule For the van der Waals equation of state: V-b(P+ a/v^2)=RT the derivative (∂V/∂T)p is difficult to obtain directly because finding an equation for V in terms...
  40. P

    Kinematics problem, don't know how to find unknown variables

    Homework Statement A rocket is shot straight up with an initial speed at 500 m/s. Homework Equations The equations the teacher went over beforehand were Yf = 1/2 A(T)^2 + Vi(T) + Yi Vf = A(T) + Vi The Attempt at a Solution So far I have Vf = (-9.8)(T) + 500 m/s Yf = 4.9T2 +...
  41. S

    Finding the curl in diffrent coordinates by transforming variables

    we have a well known and simple equation for curl in cartesian coo. now we want it in let's say cylindrical coordinates. question is...can we transform every thing to cylinderical and then use the formula for cartesian?I mean writing basis vectors of cartesian in terms of r and theta and z and...
  42. D

    How do i find the final velocity with 2 height variables and gravity

    1.A tennis ball is dropped from 1.3 m above the ground. It rebounds to a height of 0.956 m. With what velocity does it hit the ground? The acceleration of gravity is 9.8 m/s^2 (Let down be negative.) Answer in units of m/s 2.With what velocity does it leave the ground? Answer in units of m/s...
  43. S

    Combustion potato gun - Need associated variables, ideas and concepts.

    Hi Guys, I have a Year 11 Project weighted 15% of my year mark due in 5 weeks. For my project I've decided to build a classic combustion potato gun. This gun will consist of pvc piping, barrel, Lantern lighter, any other components and of course hairspray (combustion) Anyways It is still only...
  44. A

    Bernoulli Distribution/ Random Variables

    Homework Statement Take Ω = [0, 1] and P the uniform probability. (a) Give an example of two random variables X and Y defined on Ω that both have the same Bernoulli distribution with parameter θ = 1/3. (b) Give an example of such random variables that are independent, and not independent...
  45. K

    Separation of variables on 2nd order ode

    Hi all Quick one, if one had an equation y' = x on could simply separate the variables and integrate. Now it the equation y'' = x you would use separation of variables what drives this? Also y'' =0. Is the same as. y''dx =0 dx Why is this legal?Thanks in advance
  46. T

    Subtracting 2 Fractions with Variables to Exponents: Explained

    -2((1/((x^2 + 1)^2 )) - ((4x^2 )/(x^2 +1)^3 )) = 6x^2-2/(x^2+1) This is actually the last step of this problem. I understand everything they did up until here, but I'm a bit confused as to how they got from their last step, to the actual answer. Could someone explain?
  47. S

    Change of variables in a second order SDE

    Hello everyone! I am fairly new to SDE theory, so I'm sorry if my question may be a bit naive. I have the following coupled set of SDE:s d\phi = \frac{v - v_r}{R}d t + \frac{\pi}{\sqrt{t_c}}d W d v = A\cos(n\phi - \phi_w)d t + a_v d t + \sigma_v d W. W denotes a Wiener process, and the...
  48. camilus

    Top grad programs in several complex variables and complex geometry?

    I know UC San Diego is good, Rothschild and Ebenfelt are there, but are there any other ones that stand out in these and related fields? Thanks a bunch
  49. M

    Limitations of Comparing t/U and t^2/U Variables

    Let U and t be independent real variable. In the limit: t/U \rightarrow 0 Can I say that t^2/U \rightarrow 0 too? Or I can compare only same powers of both? Regards.
  50. J

    Elementary algebra of complex variables problem

    I'm having difficulty deducing that Re z = 0.
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