Derivatives and Integrals of units

[Dougggggg]

I couldn't decide whether to place this in the Physics or the Math section of the forums, deep down it is really a Math question for Physics problems. So mods please move if you feel it would be more appropriate in the Physics section.

So when doing calculations, I always like to make sure my units are behaving correctly. That I am not adding kg to m/s or something weird like that. One thing that I haven't really thought about how it applies is doing calculus type manipulation of units. Like if I take the derivative of a unit of displacement, with respect to a unit of time, I get a unit of displacement over time.

For Integrals if I take a integral of force, with respect to displacement, I get a unit of force times displacement.

Basically, what is the basic method to these kind of things?

 

[CompuChip]

It is exactly as you say.
When you differentiate with respect to x, the units get a factor of 1/[x] (where [x] are x's units).
When you integrate with respect to x, the units get a factor of [x].
For differentiation, this follows simply from the definition, because f'(x) is a limit of the quotient
$$\frac{\Delta f(x)}{\Delta x}$$
which has units
$$\frac{[\Delta f(x)]}{[\Delta x]} = \frac{[f(x)]}{[x]}$$.

For integration, simply reverse the argument, i.e. something along the lines of
$$\left[ \frac{d}{dx} \left( \int f(x) \, dx \right) \right] = [ f(x) ]$$
but it is also
$$\left[\int f(x) \, dx \right] / [x]$$
therefore,
$$\left[ \int f(x) \, dx \right] = [f(x)] [x]$$.

 

[HallsofIvy]

Think of differentiation as being an extension of division and integration as an extension of multiplication.

The units of $dy/dx$ are the units of y divided by the units of x. For example, if y is measured in meters and x is measured in seconds then dy/dx (the rate of change of y with respect to x= rate of change of distance with respect to time) has units of "meters per second".

The units of $\int f(x) dx$ are the units of f(x) times the units of dx (which are the same as the units of x). For example, if f(x) is a "linear mass density" in "kilograms per meter" and x has units of meters, then $\int f(x)dx$ has units of (kilogram/meter)(meter)= kilogram.