Variable in Math

Question

What is formal definition of "variable" ? I cannot able to understand variable because some times it is varying or some time used for approaching. Arbitrary constant also confusing me. In Multi variable calculus variable treat as constant, why? it is actual varying...

Answer 1

A variable is a quantity in a mathematical expression that can be assigned different values. For the purpose of a particular analysis, it is perfectly possible to consider that one variable, for the present, is fixed, while another is allowed to change freely. This is what is being done when taking partial derivatives.

An arbitrary constant is a quantity that is fixed for the purpose of the present analysis, but whose actual value is unknown, at least until further information is used.

Answer 2

Let me guess what is confusing you.

In one variable calculus, if $u=u(x)$, then (on some interval) $$ \frac{d}{dx}u(x)=0\quad\iff\quad u(x)=C. $$

In several variable calculus (I use two variables below), if $u=u(x,y)$, then (in some domain) $$ \frac{\partial}{\partial x}u(x,y)=0\quad\iff\quad u(x,y)=\phi(y) $$ where $\phi$ only has $y$ as a variable.

In both cases we can think of that the fact that the derivative with respect to $x$ is zero means that the function is not allowed to depend on that variable, i.e. $x$. In the first case, it has no other variables to depend on, and thus must be a constant. In the second case, there is a second variable $y$, and the fact that the $x$-derivative is zero says nothing about how the function varies in the $y$-direction. Hence, $u$ can be any function depending on $y$.