What exactly is multivariable calculus? i was learning about multi variable calculus. But, i didn't quite understand what does it mean. Let me demonstrate what i mean:
In single variable calculus, we had an input that gave us an output. For example, $y=x^2$, we have one input, i.e $x$, that gave rise to one output, i.e $y$, by squaring the $x$. So, if we had $x$ as $2$, we would have $y$ as $4$, and then we would just plot $(2,4)$ on a 2-dimensional plane. But, I am not able to think multi variable calculus as the way i just showed. I mean, are we applying 2 seperate functions to an input variable to get 2 variables, i.e, are we doing this: $(y,z) = f(x), g(x)$, or what?
 A: Standard calculus studies functions that have a single (usually real) number as input, and a single number as output. What calculus does is to differentiate and integrate these functions, and tell you what the answers mean. Multivariable calculus is the same, just with more variables.
You have functions that have multiple variables as input and a single number as output (typical real-world examples include the altitude as a function of the two coordinates on a map, or the temperature in a room as a function of the position coordinates). You can have a function that takes in a single variable as input and gives out several numbers (the position of a particle as a function of time is a common example).
And finally, you can do both and have a function that takes in multiple variables, and gives out multiple numbers (winds and currents are common real-world examples, and a very important, more abstract example is the coordinates for a point in one coordinate system given its coordinates in another coordinate system). (Note that there is, a priori, no reason why the number of input variables and number of output numbers should be the same. It just so happens in many concrete examples that they are the same.)
These functions are the base objects of study. Then we develop notions of derivatives and integrals that make sense, and that's multivariable calculus.
