# What do these symbol mean?

I always see these symbols and others like it when looking at really advanced maths. I have yet to learn anything about it. I was wondering if someone could explain briefly what they are used for.

$$\oint \quad\iint \quad \iiint$$

• Are you familiar with $\int$? – graydad May 21 '15 at 18:59
• Yes. I am familiar with the integral specifically the Riemann integral which I have seen rigorously defined in my university class. – Terry Chao May 21 '15 at 19:03
• I'd be tempted to say that the symbols need extra information. $\iint dA$ representing a double integral. $\iint dxdy$ an iterated integral. – Karl May 21 '15 at 19:07
• @Karl arguably, $\iint dA$ is ill-defined. $dA$ could be thought of as a product measure $dA = d(x,y)$ and as such should have been written as $\int dA$. See en.wikipedia.org/wiki/Fubini%27s_theorem – JMoravitz May 21 '15 at 19:15
• @JMoravitz Thanks that actually makes more sense. – Karl May 21 '15 at 19:21

The first is Contour Integral The second and third are double and triple integrals

• Not necessarily a countour integral, could be used to denote a surface/boundary integral also. – Rammus May 21 '15 at 19:03

The $\oint$ is called line integral

The terms path integral, curve integral, and curvilinear integral are also used; contour integral as well, although that is typically reserved for line integrals in the complex plane.

using it we take perimeter instead of area in contrast to $\int$. This finds great use in physics (ampere circuital law)

The $\iint$ it is the definite integral of 2 variables $f(x,y)$ can be used to take Volume of 3-D figures . There is also a line integral version of double integral called Surface integral this calculates surface area.

The $\iiint$ it is the definite integral of 3 variables $f(x,y,z)$ can be used to take volume of a 4 D figure( wow!)

• The $\rm\LaTeX$ for $\oint$ is \oint, not \lint. – Cameron Williams May 21 '15 at 19:05
• The first is an integral over a closed curve (or more generally -- some closed region). Also $\iint$ and $\iiint$ don't have to be used in $\Bbb R^3$ -- it's just that after a first course in multivariable calculus, we generally stop writing $\iint$ and $\iiint$ and just stick to $\int$ and $\oint$. – user137731 May 21 '15 at 19:06
• @Cameron thanks for that – Ilaya Raja S May 21 '15 at 19:13
• $\iint f(x,y) dxdy$ is a volume I thought. – Karl May 21 '15 at 19:15
• It depends on your perspective. What @IlayaRajaS is referring to is that $\iint_S 1 dx\,dy$ gives you the area of $S$, and $\iiint_S 1 dx\,dy\,dz$ gives you the volume of $S$. – Alex Kruckman May 21 '15 at 19:22

The first denotes a closed integral. Its meaning varies a little based on context, but the best way to think of it is that the set of points over which you are integrating is closed and bounded (like integrating over the circumference of a circle or the outside of a sphere). There might be a little bit of contention with the leftmost one since physicists kind of co-opted the notation. A mathematician might be tempted to say that it is a closed contour or line integral (integrating over a closed curve) but in physics, the notation is also used for flux integrals like in Maxwell's equations. The only way to be $100\%$ sure what is meant is to look at what kind of infinitesimal notation is used ($dl$ versus $dS$).

The latter two have very specific meaning: the middle symbol denotes a double integral, i.e. there are two integration variables, and the rightmost symbol denotes a triple integral, i.e. there are three integration variables.

• What does having three or even two variables in the integration mean? – Terry Chao May 21 '15 at 19:05
• @TerryChao Just like integrating over one variable gives area, integrating over two gives volume. Think of it like having Riemann rectangular prisms instead of rectangles. There isn't as obvious of an interpretation for triple integrals outside of specific instances. – Cameron Williams May 21 '15 at 19:06
• Okay thanks, I shall be learning more about these next year I believe thanks for this comments. – Terry Chao May 21 '15 at 19:08