Recently in my mathematics courses I was taught partial derivatives, and I wondered if the reverse exists for integrals.
This may sound like a stupid question, and it probably is, but let me explain:
By the fundamental theorem of calculus:
$$ \int \frac{d}{dx}f(x)~dx = f(x) $$
So is there an operator such that:
$$ \int \frac{\partial}{\partial x}f(x,y)~\partial x = f(x,y) $$
or is this complete bogus?
What I am saying is that if the partial derivatives of a multi-variable equation is the slope of the line along the derivative axis, is there an operator, a "partial integral" that is the area under the curve along the integrated axis?
Or is that just $\int f(x,y) ~dx$, since doing it over more than one axis requires $\iint f(x,y) dx dy$?
Physically, this could be related to finding the x-component of the velocity given the acceleration function.
Also, how stupid of a question is this?
Best Answer
Your partial integral is roughly the same as your regular integral, with a caveat. If you have, say, $$\int \frac{d}{dx} f(x) dx$$ When you integrate this you end up with $f(x) + C$ - since this is the antiderivative of $f'(x)$, the $C$ shows up because integration only knows 'so much' - the derivative of $C$ is zero, so we don't know whether or not it's actually in $f(x)$. Similarly, when we take an integral over one variable, we get $$\int \frac{\partial}{\partial x} f(x,y) dx$$ The partial 'knocks out' any functions of $y$ in $f(x,y)$; for example, if $f(x,y)=xy+y^2$, then the partial will send $y^2$ to zero. So as before, when we integrate solely with respect to $x$ of a multivariable function, we get $$\int \frac{\partial}{\partial x} f(x,y) dx = g(x,y)+C(y) =f(x,y)$$ Where $C(y)$ denotes any function of $y$. There's no way to get an integral that will 'invert' the partial operator while still knowing about $y$ - that information is lost when we took the partial in the first place.