The aim of this very short note it to show that even in

a totally elementary framework, it is possible to glimpse the Stokes

theorem

Goldoni, Luca

Unitn-eprints Research

LOOKING FOR STOKES THEOREM IN ANELEMENTARY TRAPEZOID.LUCA GOLDONIAbstract. The aim of this very short note it to show that even ina totally elementary framework, it is possible to glimpse the Stokestheorem.The NoteFigure 1The Stokes theorem tell us that(1)∫Ωdω =∫∂Ωω.where Ω is a suitable domain and ∂Ω its boundary. Even in the calcu-lation of the area of an elementary trapezoid, we use, almost certainlyin a non-conscious way, either the LHS or the RHS of equation (1). LetDate: September 21, 2012.1991 Mathematics Subject Classification. 97A80,97A99.Key words and phrases. elementary mathematics, Stokes Theorem,Dipartimento di Matematica. Universita` di Trento.12 LUCA GOLDONIus consider a rectangular trapezoid ABCD as shown in Figure 1 andlet be y = mx the straight line through O and C. Of course we cancalculate the area of the trapezoid in two different ways:(1) We can think to the trapezoid as equivalent to the rectangleABIH .(2) We can think to the trapezoid as the difference between thetriangles COB and DOA.If we think to the trapezoid as equivalent to the rectangle ABIH then,since its area is A = y(b− a), we can imagine this as theas the LHS of(1) where Ω is the interval [a, b] and dω = (mx)dx. If we calculate thearea as difference between the area of the triangle OCB and the areaof the triangle DOA then we haveA =b (mb)2−a (ma)2=12mb2 −12ma2 = ω (b)− ω (a) =∫∂Ωωnamely, the RHS of (1) because the boundary of Ω, trivially is givenby {b, a}.Universita` di Trento, Dipartimento di Matematica, v. Sommarive14, 56100 Trento, ItalyE-mail address : goldoni@science.unitn.it

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Looking for the Stokes Theorem in an elementary trapezoid

University of Trento : UNITN-eprints

LOOKING FOR STOKES THEOREM IN AN
ELEMENTARY TRAPEZOID.
LUCA GOLDONI
Abstract. The aim of this very short note it to show that even in
a totally elementary framework, it is possible to glimpse the Stokes
theorem.
The Note
Figure 1
The Stokes theorem tell us that
(1)
∫
Ω
dω =
∫
∂Ω
ω.
where Ω is a suitable domain and ∂Ω its boundary. Even in the calcu-
lation of the area of an elementary trapezoid, we use, almost certainly
in a non-conscious way, either the LHS or the RHS of equation (1). Let
Date: September 21, 2012.
1991 Mathematics Subject Classification. 97A80,97A99.
Key words and phrases. elementary mathematics, Stokes Theorem,
Dipartimento di Matematica. Universita` di Trento.
1
2 LUCA GOLDONI
us consider a rectangular trapezoid ABCD as shown in Figure 1 and
let be y = mx the straight line through O and C. Of course we can
calculate the area of the trapezoid in two different ways:
(1) We can think to the trapezoid as equivalent to the rectangle
ABIH .
(2) We can think to the trapezoid as the difference between the
triangles COB and DOA.
If we think to the trapezoid as equivalent to the rectangle ABIH then,
since its area is A = y(b− a), we can imagine this as theas the LHS of
(1) where Ω is the interval [a, b] and dω = (mx)dx. If we calculate the
area as difference between the area of the triangle OCB and the area
of the triangle DOA then we have
A =
b (mb)
2
−
a (ma)
2
=
1
2
mb2 −
1
2
ma2 = ω (b)− ω (a) =
∫
∂Ω
ω
namely, the RHS of (1) because the boundary of Ω, trivially is given
by {b, a}.
Universita` di Trento, Dipartimento di Matematica, v. Sommarive
14, 56100 Trento, Italy
E-mail address : goldoni@science.unitn.it


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Looking for the Stokes Theorem in an elementary trapezoid

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