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I am currently taking Calc 3 and we just finished our unit on double/triple integrals. I started thinking about a problem back from AP Physics (where my teacher did an impressive amount of hand waving to somehow avoid directly explaining vector calculus when discussing Maxwell's equations, leading to me spending an entire semester confused about what the heck flux was supposed to be or why it should matter), which basically boils down to $\iiint_D{F(x, y, z)}{dV}=Q(D)$. In Calc 1 and 2, if you have an equation of the form $\int{f(x)}{dx}=g(x)$, then you can take the "inverse integral" (i.e. derivative with respect to x) of both sides, which gives $f(x)=\frac{d}{dx}g(x)$. However, I cannot seem to figure out what the analogue of this would be for a double or triple integral. What is the opposite/inverse operation for a double or triple integral over some domain $D$?

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    $\begingroup$ There isn't any, this is specific to one dimension only $\endgroup$ – Ninad Munshi Oct 22 '20 at 8:21
  • $\begingroup$ @NinadMunshi Shouldn't there at least be some way to define a new operation such that it is the inverse of a double integral over some particular region D? Even if it isn't an analytic operation, it still should exist. Much like the function xe^x doesn't have an algebraic inverse, but we can still define a non-elementary function W(x) such that it has the property W(xe^x) = x for all real numbers x. $\endgroup$ – Null Spark Oct 22 '20 at 8:32
  • $\begingroup$ Both $\int_0^1\mathrm dx=1$ and $\int_0^12x\,\mathrm dx=1$. Are you suggesting there should be some way of recovering the integrand of a definite integral given its value? $\endgroup$ – user170231 Oct 22 '20 at 16:56
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It's convenient but misleading to write $\int f(x) \, dx = g(x)$. The RHS is a function of $x$ but the LHS is not until you write down some bounds of integration, and once you do the inner $x$ has been integrated over so it's a "dummy variable" that has nothing to do with the $x$ on the RHS. Less ambiguous is to write

$$\int_a^x f(t) \, dt = g(x)$$

which makes the dependence on $x$ of the LHS clear and emphasizes that the variable being integrated over is a dummy variable.

This is not just pedantry; the point of doing this is to emphasize what changes when you move from integrals to double integrals. To think of a double integral as a function of two variables, rather than just as a function of a region being integrated over, we need to pick regions to integrate over that are defined by two parameters. An easy choice is a rectangle $[a, x] \times [b, y]$, which lets us write expressions like

$$\int_{t=b}^y \int_{s=a}^x f(s, t) \, ds \, dt = g(x, y)$$

and if we set things up like this we can recover $f$ by differentiating twice, using the fundamental theorem of calculus twice:

$$f(x, y) = \frac{\partial}{\partial x} \frac{\partial}{\partial y} g(x, y).$$

This generalizes to $n$ dimensions too. But it's worth emphasizing that in order to do this we had to choose to integrate over rectangles, and there are lots of other choices of how to parameterize regions to integrate over that we could make instead.

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  • $\begingroup$ Ahh, I see. If you're given any particular domain D to take a double integral over, is there a way to find a corresponding inverse then? For instance, say that D is the disk {(x, y) in R^2 | x^2 + y^2 <= 1}. $\endgroup$ – Null Spark Oct 22 '20 at 8:35
  • $\begingroup$ @Null: if you fix the domain then the output of integration over that domain is just a number. That's not enough information to recover a function from. This is already true in one dimension. $\endgroup$ – Qiaochu Yuan Oct 22 '20 at 8:39
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    $\begingroup$ @Null: that's a very different question. Maybe the simplest "physical" interpretation is that $F$ describes a "density" and the triple integral describes a "total mass." This interpretation only makes sense if $F$ is non-negative, though. $\endgroup$ – Qiaochu Yuan Oct 22 '20 at 9:02
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    $\begingroup$ Although I personally remember that when I was just learning calculus for the first time, I was very confused by the fact that the "$x$"'s on the two sides are conceptually very different, and I think that the more pedantic notation might have been pedagogically useful. This is pretty clear when you expand out (one possible) full definition of the derivative: $f'(x_0) := \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}.$ $\endgroup$ – tparker Oct 26 '20 at 3:00
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    $\begingroup$ Similarly, the standard notation leads to ambiguous questions like "If $f(x) = x^2$ and $x = 5$ then what is $df/dx$?" It's clear to an expert what question is being asked here in a pedagogical context, but it's understandable for a beginner to reason that "$f(x) = 25$ so $df/dx = d/dx\ 25 = 0$ so the answer is 0." Using two different variables would help eliminate this ambiguity. $\endgroup$ – tparker Oct 26 '20 at 3:08
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To complement Qiaochu Y. excellent explanation note that, same as we write $$ \eqalign{ & \int {f(x)dx} = g(x) = \int_{t = a}^x {f(t)dt} + c = \int\limits_{t \in \left[ {a,x} \right]} {f(t)dt} + c\quad \Rightarrow \quad \cr & \Rightarrow \quad d\int_{t = a}^x {f(t)dt} = \int\limits_{t \in \left[ {x,x + dx} \right]} {f(t)dt} = f(x)dx = {d \over {dx}}g(x)dx \cr} $$ with all cautions as for continuity, and with the algebraic meaning for negative $dx$,

then, specially in physics, we often write $$ \eqalign{ & \int\!\!\!\int\limits_A {f(x,y)dxdy} = \int\!\!\!\int\limits_A {f(x,y)\delta A} = g(A)\quad \Rightarrow \cr & \Rightarrow \quad \delta \int\!\!\!\int\limits_A {f(x,y)\delta A} = \int\!\!\!\int\limits_{\delta A} {f(x,y)dxdy} = {d \over {dA}}g(A)\,\delta A \cr} $$ where the $\delta A$ is also taken with the algebraic sign.

Delta_Int_Mult_1

From here we reach to Stoke's Theorem.

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