darkdave3000 said:
I have a question about math function notation.
f(x) = a function of x
f(x,y) = dy/dx <---- Is this true?
If you mean to say that "f(x) is a function of x" is given information in some situation then you are correct to object to the use of "f(x,y)" to denote a function of two real variables in the same situation. (However, such ambiguous notation is sometimes used.)
If you mean to say that , by tradition, "f" is always used as the name of a function of one real variable, that, of course, is not correct. It also
not correct to think that if "y" is used to denote an "unknown function of x" that we must use the notation "y = f(x)" and name that unknown function "f".You will find that people writing about applications of mathematics , especially applications of mathematics to physics, use notation and terminology that is technically wrong because it is ambiguous. Ambiguous notation is convenient, but it can also be confusing.
For example, let ##f(x,y)## be a function of two variables defined by ##f(x,y) = x + xy ##. The function has a domain consisting of pairs of real numbers. However, a book that uses that notation will often introduce other information without changing notation. For example, suppose we add the information ##y = 3x##. It is very tempting to continue to use the name "##f##" to denote the function of one real variable given by ## x + x(3x)##. Technically, this is incorrect notation because it is an ambiguous use of the name "##f##". Nevertheless, in such situation a book might use notation like ##\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}##, where "##f##" on the left hand side of the equation denotes a function of a single variable and "##f##" on the right hand side denotes a function of two real variables.
f(x,y) = f ' (x) ? Is this true also? I've been confused by this logic when I watch youtube videos about computational math such as Euler method and Runge Kutta method
In the material you are watching, I think the idea might be ##f(x,y) = \frac{dy}{dx} = y'(x) ## The "unknown function" is named "##y##", not "##f##".
A typical scenario for differential equations is that "##y##" denotes a ( perhaps unknown) function of a single real variable. We could denote this function as "##y(x)##". However, that notation would get confusing because when we wish to denote the function "##y##" times an algebraic expression, the "##( )##" could be mistaken for multiplication. For example, to denote the concept "The function ##y## evaluated at the value ##t## times the expression ##(t+3)## evaluated at that same value", such notation would give us ##y(t)(t+3)## which might be mistakenly manipulated into ##y(t^2 + 3t)##.
To define a differential equation, we can begin with a real valued function ##f## of several real variables. For example let ##f(a,b,c) = c + ab + 3##, which is a function of 3 real variables. Using the definition of ##f##, we write can an
equation ##f(a,b,c) = 0## (Don't confuse the concept of an
equation with the concept of a
function.) The differential equation is defined using ""##y##" to denote an unknown function and "##y'##" to denote the derivative of that function. A solution to the differential equation "##f(x,y,y') = 0##" is defined to be a function ##y## of a single real variable such that when ##y## is evaluated at the value ##x##, we have ##f(x,y(x),y'(x)) = 0## In the particular case of ##f(a,b,c) = c + ab + 3##, the differential equation would be ##y'(x) + xy(x) + 3 = 0##. However, as mentioned above, the argument "##(x)##" is traditionally omitted when we write the function ##y##, so the equation would be written ## y' + xy + 3 = 0## or ##\frac{dy}{dx} + xy + 3 = 0 ##
Notation in applied mathematics shows no mercy to people who do not understand the concept of the "scope" (i.e. the context) of variables. In the above example, I used the notation ##f(a,b,c) = c + ab + 3##" in a effort to be clear. However, it would be permissible to define ##f## by the notation "##f(x,y,z) = z + xy + 3##" or even (to be confusing) as "##f(z,y,x) = x + zy + 3##", because variable names are arbitrary within a given "scope" of the variables. The names "##x##" and "##y##" used in the scope for defining ##f## can be unrelated to the use of those names when they appear farther down the page in the discussion of "##y##" being a function of "##x##".
