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I want to know whether "partial integration" exists analogous to partial differentiation in ordinary calculus for functions of several variables.

Jonas Meyer
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GAURAV SHARMA
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    partial integration is synonym for integration by parts – Pedja Nov 07 '11 at 16:59
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    If I understand the question correctly, I'd say yes. For example, you can compute $\int x^2+3y^2 , dy= yx^2+y^3 +C(x)$ (the constant of integration becomes an arbitrary function of $x$). – David Mitra Nov 07 '11 at 17:03
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    I think you're looking for line integrals, possibly along a coordinate line. – hmakholm left over Monica Nov 07 '11 at 17:04
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    @pedja No, partial derivatives are looking at the change in only one coordinate. Integration by parts is undoing the product rule. – GeoffDS Nov 07 '11 at 17:06
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    @Graphth partial integration – Pedja Nov 07 '11 at 17:11
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    @pedja I see, but the point of the question is to find an integration that is analogous to partial derivatives. Is integration by parts analagous to partial derivatives? I don't think so. So, it's just that the OP doesn't know the correct term. Your comment is saying that term already exists but doesn't describe what the OP is asking about? – GeoffDS Nov 07 '11 at 17:36
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    @Graphth,I just pointed out on wrong use of the term – Pedja Nov 07 '11 at 17:44
  • @pedja Okay, sorry, I misunderstood. – GeoffDS Nov 07 '11 at 18:51
  • @pedja To the best of my knowledge, "partial integration" used to refer to integration by parts is just a mistranslation of the German term and as a calque for other foreign terms. In my American courses "partial integration" and "integration by parts" have always been distinct concepts, the former actually being an inverse operation to partial differentiation. – Chill2Macht Nov 05 '16 at 08:42

1 Answers1

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First, please do note as pedja has mentioned in the comments above that the term "partial integration" is synonymous with "integration by parts".

What you're looking for might better be called something like "partial antiderivatives". These things do show up frequently. Although I don't think they really have a definite name. Here are a couple of examples.

Suppose we wish to compute a double integral $\iint_R x^2y\,dA$ where $R$ is the rectangle $[0,1]\times [0,2]$. Then Fubini's theorem tells us that the double integral can be computed via iterated single integrals as follows:

$$ \iint_R x^2y\;dA = \int_0^1 \int_0^2 x^2y \;dy\;dx = \int_0^1 \left[ \frac{1}{2}x^2y^2 \right]_0^2 \;dx = \int_0^1 2x^2 \,dx = \left. \frac{2}{3}x^3 \right|_0^1 = \frac{2}{3} $$

Let's focus on the inner integral. I used the fact that $x^2y$ when holding $x$ constant and integrating with respect to $y$ is $\frac{1}{2}x^2y^2 + C(x)$. When plugging in endpoints the "$C(x)$" term disappears (just as the constant does not affect the value of a single definite integral). In some sense you could consider $\frac{1}{2}x^2y^2+C(x)$ the partial integral or partial antiderivative of $x^2y$ with respect to $y$. Although I don't think anyone uses such terminology. Also, notice that we have a whole function of $x$ instead of a constant because partial differentiation with respect to $y$ kills such things.

Another place this sort of thing shows up is when one finds potential functions for conservative vector fields. Let ${\bf F}(x,y)=\langle 3x^2+y, x+5 \rangle$.

We integrate the first component with respect to $x$ and the second component with respect to $y$ (these are your "partial" integrals):

$\int 3x^2+y\;dx = x^3+xy+C_1(y)$ and $\int x+5\;dy = xy+5y+C_2(x)$. We want a function $f(x,y)$ such that $f(x,y)=x^3+xy+C_1(y)=xy+5y+C_2(x)$. The only way to reconcile these expressions is to let $C_1(y)=5y+$constant and $C_2(x)=x^3+$constant. So if $C$ is a constant, we have $f(x,y)=xy+x^3+5y+C$ where $\nabla f={\bf F}$.

Bill Cook
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