Function notation in physics equations

In summary, the conversation discusses the use of function notation in physics equations and raises questions about the distinction between a variable and a function. It is noted that there is no strict rule for designating a function and any letter can be used as long as it is defined clearly. The conversation also touches on the ambiguity of what is considered a function in physics.
  • #1
Mr Davis 97
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I have a question about function notation as it is used in physics equations.

Say that we have the equation

$$x = x_{0} + v_{0}t + \frac{1}{2}at^{2}$$

Obviously, x is a variable quantity representing all possible displacement values that are dependent on the value t. But, what happens when we do

$$x(t) = x_{0} + v_{0}t + \frac{1}{2}at^{2}$$

Does x then become a function rather than a variable? Typically, in function notation, we denote f as the function and f(x) as the variable output that depends on x.

So in the case of the above equation, how do we reconcile that x constantly changes from a variable quantity to the name of a function?
 
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  • #2
In both cases, x is a dependent variable. In the first case, it's implied to be t and in the second its explicitly stated.
 
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  • #3
jedishrfu said:
In both cases, x is a dependent variable. In the first case, it's implied to be t and in the second its explicitly stated.
To clarify a bit what jedi said, in the first equation, it is implied that t is the independent variable, while this explicit in the second equation.
 
  • #4
Mark44 said:
To clarify a bit what jedi said, in the first equation, it is implied that t is the independent variable, while this explicit in the second equation.

Doesn't this contradict what function notation usually says though, that in f(x) f is the symbolic representation of a function, which means it's not a dependent variable. Do we say that " in x(t), x is the dependent variable" just by convention?
 
  • #5
  • #6
Bystander said:
Just googled SIAM (Society of Industrial and Applied Mathematics) and mathematical notation on the suspicion that this has been covered somewhere a la IUPAC an IUPAP, and got
http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Mathematics
which might help resolve some of the questions/issues.

I didn't find anything in the article regarding function notation.
 
  • #7
Copy this, paste that. Too many bells & whistles in my browser. http://en.wikipedia.org/wiki/Function_(mathematics) Was what I meant to paste, but going over it just now, it does NOT draw on SIAM. Speed read both, conflated SIAM reference from what just disappointed you with content of this second. There may not be any more convention on this than there is on IUPAC and naming organic compounds.
 
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  • #8
Mr Davis 97 said:
I have a question about function notation as it is used in physics equations.

Say that we have the equation

$$x = x_{0} + v_{0}t + \frac{1}{2}at^{2}$$

Obviously, x is a variable quantity representing all possible displacement values that are dependent on the value t. But, what happens when we do

$$x(t) = x_{0} + v_{0}t + \frac{1}{2}at^{2}$$

Does x then become a function rather than a variable? Typically, in function notation, we denote f as the function and f(x) as the variable output that depends on x.

So in the case of the above equation, how do we reconcile that x constantly changes from a variable quantity to the name of a function?
You are correct. An extremely pedantic mathematician would write x = f(t) and then define f to be the function defined by [itex]f(t) = x_{0} + v_{0}t + \frac{1}{2}at^{2}[/itex], for all [itex]t\geq 0[/itex], where a function is a rule or set. However, such abstraction is really unnecessary at this level of physics, so it is common to conflate variables with function names. Thus we get notation like x = x(t), where the x on the left side is a variable, an element of the image of a function, while x on the right side is a function, a set or rule.
Physicists tend to be rather flippant in notation, so you can expect more conflation in the future. Mathematicians tend to inspect the structure of operations in more detail than would necessarily arise in a physical application, which requires the more precise language that separates the value of a function from a function. As such, expect vectors to be conflated with vector fields, and likewise for tensors and tensor fields.
 
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  • #9
Only mathematicians really care about the correctness of notation. In the case you presented,

##x(t) = \frac{1}{2}at^2 + v_0 t + x_0##

##x_0## is just the initial value for the function, just like you might write a linear equation like:

##y(x) = ax + y_0##

Though I don't know how many people actually do that; but ##x## is the function value at time ##t##, so ##x_0## is like saying, "##x(0)##", if you like. You could replace it by anything sensible, like "##s_0##", or "##p_0##". Just as long as you don't think that ##x## is the independent variable.
 
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  • #10
There's no hard and fast rule, in my experience, which states that functions must only be designated using an 'f', e.g., f(x), f(t), etc.

Working with polar coordinates and vectors, often the position function is designated r(θ) or r(t), for example.

In working with electricity, the current can be a function of time, and its function is designated as i(t); similarly, the voltage function can be designated v(t).

In working with the structural mechanics of beams, the shear force, bending moment, slope, and deflection functions w.r.t. position are indicated V(x), M(x), Θ(x), and δ(x), respectively, quite frequently.

As long as the notation is explicitly defined and clear to the reader, you can use whatever letter to indicate a function, as long as it is distinct from any independent variables on which it depends.
 
  • #11
In physics, what is a function of what is often hard to determine. For example, suppose a an object of given mass is initially at rest at height h near the surface of the Earth and it is dropped and falls. During the fall, if given its momentum you could find its position. So in that situation, to a mathematician, position is a function of momentum. Given such cases exist, if you had a function F of position and momentum and took its derivative with respect to momentum, a mathematician would include a term with the partial derivative of F with respect to position times the partial derivative of position with respect to momentum. I don't recall seeing physicists do that.
 

1. What is function notation in physics equations?

Function notation in physics equations is a way of representing a mathematical relationship between two variables. It is commonly used in physics to describe how one variable, such as position or velocity, changes in relation to another variable, such as time.

2. How is function notation written?

Function notation is written as f(x), where f is the name of the function and x is the independent variable. For example, the function representing the position of an object over time would be written as f(t), with t representing time.

3. What is the purpose of using function notation in physics equations?

The purpose of using function notation in physics equations is to make it easier to understand and analyze the relationship between variables. It allows for a more concise and organized way of representing complex equations and allows for easy substitution of values to solve for unknown variables.

4. How is function notation used in physics problem solving?

In physics problem solving, function notation is used to describe the relationship between variables in an equation and to solve for unknown variables by substituting known values for the independent variable. It also allows for the use of calculus to find rates of change and optimize functions.

5. Are there any common mistakes when using function notation in physics equations?

One common mistake when using function notation in physics equations is forgetting to properly label the independent and dependent variables. It is important to correctly identify which variable is being manipulated and which variable is being affected by the manipulation in order to accurately use function notation. Another mistake is not properly substituting in values for the independent variable, which can lead to incorrect solutions.

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