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Example of the?????fonts.

Paul Pichaureau

August 29, 2013

Abstract

The package?????consists of a full set of mathematical fonts, de- signed to be combined with Bitstream Bitstream Charter as the main text font. This example is extracted from the excellent bookMathematics for Physics and Physicists, W. APPEL, Princeton University Press, 2007.

1 Conformal maps

1.1 Preliminaries

Consider a change of variable(x,y)7!(u,v) =u(x,y),v(x,y)in the plane R

2, identified withR. This change of variable really only deserves the name

iffis locally bijective (i.e., one-to-one); this is the case if the jacobian of the map is nonzero (then so is the jacobian of the inverse map):

D(u,v)D(x,y)

@u@x@u@y @v@x@v@y

6=0 andD(x,y)D(u,v)

@x@u@x@v @y@u@y@v 6=0.

Theorem 1.1.In a complex change of variable

z=x+iy7!w=f(z) =u+iv, andiffis holomorphic, then the jacobian of the map is equal to J f(z) =D(u,v)D(x,y) =f0(z)2. 1 Dem.Indeed, we havef0(z) =@u@x+i@v@xand hence, by the Cauchy-Riemann relations, 2 2 =@u@x@v@y@v@x@u@y=Jf(z).Definition 1.1.Aconformalmaporconformaltransformationofanopensubset ΩR2into another open subsetΩ0R2is any map f:Ω7!Ω0, locally bijective, that preserves angles and orientation. Theorem 1.2.Any conformal map is given by a holomorphic function f such that the derivative of f does not vanish.

This justifies the next definition:

Definition 1.2.Aconformaltransformationorconformalmapofanopensubset ΩRinto another open subsetΩ0Ris any holomorphic function f:Ω7!Ω0 such that f

0(z)6=0for all z2Ω.

Dem.[that the definitions are equivalent]We will denote in generalw=f(z). Consider, in the complex plane, two line segmentsγ1andγ2contained inside the setΩwherefis defined, and intersecting at a pointz0inΩ. Denote byγ0

1andγ0

2their images byf.

We want to show that if the angle betweenγ1andγ2is equal toθ, then the same holds for their images, which means that the angle between the tangent lines toγ0

1andγ0

2atw0=f(z0)is also equal toθ.

Consider a pointz2γ1close toz0. Its imagew=f(z)satisfies lim z!z0ww0zz0=f0(z0), and hence limz!z0Arg(ww0)Arg(zz0) =Argf0(z0), which shows that the angle between the curveγ0

1and the real axis is equal to

the angle between the original segmentγ1and the real axis, plus the angle α=Argf0(z0)(which is well defined becausef0(z)6=0).

Similarly, the angle between the image curveγ0

2and the real axis is equal

to that between the segmentγ2and the real axis, plus the sameα. 2 Therefore, the angle between the two image curves is the same as that between the two line segments, namely,θ. Another way to see this is as follows: the tangent vectors of the curves are transformed according to the rule!V0=dfz0!V. But the differential off(when fis seen as a map fromR2toR2) is of the form dfz0=0 B

B@@P@x@P@y

@Q@x@Q@y1 C sinαcosα‹ , (1) whereαis the argument off0(z0). This is the matrix of a rotation composed with a homothety, that is, a similitude. Conversely, iffis a map which isR2-differentiable and preserves angles, then at any point dfis an endomorphism ofR2which preserves angles. Since falso preserves orientation, its determinant is positive, so dfis a similitude, and its matrix is exactly as in equation (1). The Cauchy-Riemann equations

are immediate consequences.Rem.Anantiholomorphicmap also preserves angles, but it reverses the orientation.

3

Calcul différentiel

Pour obtenir la différentielle totale de cette expression, considérée comme fonction dex,y, ..., donnons àx,y, ... des accroissementsdx,dy, .... Soient Δu,Δv, ...,Δfles accroissements correspondants deu,v, ...,f. On aura

Δf=@f@uΔu+@f@vΔv+...+RΔu+R1Δv+...,

R,R1, ... tendant vers zéro avecΔu,Δv, ....

Mais on a, d"autre part,

Δu=@u@xdx++@u@yΔy+...+SΔx+S1Δy+...

=du+Sdx+S1dy+...

Δv=@v@xdx++@v@yΔy+...+TΔx+T1Δy+...

=dv+Tdx+T1dy+... S,S1, ...,T,T1,... tendant vers zéro avecdx,dy, .... Substituant ces valeurs dans l"expression deΔf, il vient

Δf=@f@udu+@f@vdv+...+ρdx+ρ1dy+...

dx dy +...+ρdx+ρ1dy+... ρ,ρ1, ... tendant vers zéro avecdx,dy, ....

On aura donc

@f@x=@f@u@u@x+@f@v@v@x+..., @f@y=@f@u@u@y+@f@v@v@y+..., 4 et, d"autre part, df=@f@udu+@f@vdv+...; d"où les deux propositions suivantes : La dérivée, par rapport à une variable indépendante x, d"une fonction com- posée f(u,v,...)s"obtient en ajoutant ensemble les dérivées partielles@f@u,@f@v, ..., respectivement multipliées par les dérivées de u, v, ... par rapport à x. La différentielle totale df s"exprimer au moyen de u, v, ..., du, dv, ..., de la même manière que si u, v, ... étaient des variables indépendantes. CAMILLEJORDAN,Cours d"analyse de l"École polytechnique 5quotesdbs_dbs17.pdfusesText_23

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