double integral in a sentence

Examples

1. My problem is that I haven't been able to evaluate \, \ mu _ 2, and I suspect I've misunderstood something, possibly something pretty elementary about double integrals ( I have no mathematical training beyond this, and it's a " long " time ago ).
2. Discrete Green's Theorem, which gives the relationship between a double integral of a function around a simple closed rectangular curve " C " and a linear combination of the antiderivative's values at corner points along the edge of the curve, allows fast calculation of sums of values in rectangular domains.
3. Taking into account uniform conditions along the probe surface ( boundaries are excluded ), \ Delta S \ rightarrow S _ z, we can take double integral with respect to the angle \ vartheta, and with respect to the velocity v, from the expression ( 1 ), after substitution Eq . ( 2 ) in it, to calculate a total electron current on the probe
4. The above theorem was first proposed by Euler when he developed the notion of double integrals in 1769 . Although generalized to triple integrals by Lagrange in 1773, and used by Legendre, Laplace, Gauss, and first generalized to " n " variables by Mikhail Ostrogradski in 1836, it resisted a fully rigorous formal proof for a surprisingly long time, and was first satisfactorily resolved 125 years later, by �lie Cartan in a series of papers beginning in the mid-1890s (; ).
5. Note that \ iint ( the double integral ) and \ iiint ( the triple integral ) are still not kerned as they should preferably be, and are currently rendered as if they were successive \ int symbols; this is not a major problem for reading the formulas, even if the integral symbols before the last one do not have bounds, so it's best to avoid backspacing " hacks " as they may be inconsistent with a possible future better implementation of integrals symbols ( with more precisely computed kerning positions ).