physicists infinitesimals versus mathematicians differentials

7/26/2019 Physicists Infinitesimals Versus Mathematicians Differentials

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Physicists infinitesimals versus mathematiciansdifferentials

J. Tolksdorf

Max-Planck Institute of Mathematics in the Sciences

Leipzig

November 21, 2013

In the sequel, let (a, b) R be an open interval and

f : R

x y= f(x) (1)

be a differentiable function.

Using Leibnizs notation

dy

dx f

(x) (2)

for the derivative offatx , it is common in physics to interpret the left-hand side of

the above identity as thequotient of infinitesimals. Further, by formal multiplication

of both sides with dx, one gets

dy = f(x)dx . (3)

Usually physicists interpret this by saying that the physical quantity y only changes

infinitesimally with the infinitesimal change of the physical quantity x. The fol-lowing remarks are intended to somewhat clarify this use of terminology.

To have a specific physical example in mind we consider the physical analogue of the

above Leibniz form dy/dx of the derivative f(x) of a function y = f(x) with respect

to the notion of an electromagnetic current.

The notion of an (instantaneous) electromagnetic current I is defined by physi-

cists as the quotient of an infinitesimal amount of charge dq moving during an in-

finitesimal time lapse dt. This is usually expressed as

dqdt

=I . (4)


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Like any physical quantity, the electric chargeqis a quantity which carries a physicalunit (in SI-units this is the Coulomb). This unit is defined by a procedure that fixes

a typical scale. Let the latter be denoted by Q, i. e. q/Q R.

Also, the notion of time t can be made precise only with respect to a fixed time

scale T, say. The corresponding SI-unit is called the second.

Both units allow to introduce dimensionless quantities:

x:= t/T , y:=q/Q R . (5)

From this one may also introduce the dimensionless quantity A := I /IoR

, whereIo := Q/T >0 is the standard of current (SI-unit is the Ampere).

With respect to these dimensionless quantities, the above expression (4) of an elec-

tromagnetic current then mathematically expresses the assumption that there is a

differentiable function f, such that

I=f(x)Io. (6)

Let, respectively

pr1: R2

R(x, y) x (7)

and

pr2: R2 R

(x, y) y (8)

be the linear projections onto the first and second variable. The corresponding differ-

entials are usually denoted by dxand dy, i.e.

dx Dpr1(x, y) (R2) ,

dy Dpr2(x, y) (R2) . (9)

They build the dual of the standard basis on R2.

Also, let

: R2

x (x, f(x)) (10)

be the curve that is defined by the graph of the function f, i.e.

() = {(x, y) R2 | x , y= f(x)} R2 . (11)


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For every fixed x , the differential of is given by the linear mapping:

D(x) : R R2

h (h, f(x)h) . (12)

Let us call in mind that the differentiability of the function f actually means the

existence of a mapping:

A: End(R) R , (13)

such that for all h Rwith x+h :

f(x+h) =f(x) +A(x+h)h , (14)

whereby A is continuous at x . In this case, one defines the differential off at

x as

Df(x) := limh0

A(x+h) . (15)

Note that (15) means that for all 0 < |h| <

A(x+h)h= Df(x)h+o(h) , (16)

Also note that Df(x) = f(x)dx End(R), where dx is the differential of the

identity on R (i. e. the dual of 1 R). In particular,Df(x)1 = f(x) R is but the

ordinary derivative off at x . Whence, fis differentiable in x if and only if

f(x+h) =f(x) +f(x)h+o(h) (h 0) . (17)

Every physical system, which evolves in time, has a characteristic time scale , say.

Let

h:= /T


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Further, the relative quantities:

dt

T =

T = h , (21)

dq

Q =

I

Q =

I

Ioh (22)

may be regarded as being infinitesimal with respect to the considered time-scale, for

h


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With respect to the identification f(x) q(xT)/Q, it follows that

q(t+dt) =q+Idt+o(dt/T) (27)

is indeed equivalent to (14):

qt+dtT

T

Q =

q((x+h)T)

Q

= q

Q+

I

Q

dt

TT+o(dt/T)

= y+ IIo

h+o(h)

f(x) +f(x)h+o(h)

= f(x+h) . (28)

Furthermore, for q q(t+dt) q, one obtains

q

Q =O(h) (h 0) . (29)

In physicists terms:

qhQ


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Mathematics Physics

y =f(x) q =f(x)Q

dy = Df(x)= f(x)dx

dq = QDf(x)h= Iof

(x)dt= Idt

(33)

Once more, this indicates the following correspondence already mentioned:

x = t/T , y = q/Q,

dx = dt/T = h , dy = dq/Q h , (34)

which apparently looks suggestive and very familiar. Yet, the correspondence is only

formal, for the physicists infinitesimal dt = < < T is not the differential of the

physics symbol of time t as opposed to the mathematicians dx, which is the differential

of the identity on R (or the differential of the projection pr1 : R2 R onto the first

coordinate: dx = Dpr1(x, y)). Similarly, the physicists infinitesimal dq >> Q is not

the differential of the physics symbol of charge qas opposed to the mathematicians

differential dy=Dpr2(x, y).

Nonetheless, using the above (formal) correspondence, it is very tempting to write

dy

dx=

dq

Q

dtT

=dq

dtQ

T

= I

Io=f(x) , (35)

to obtain what Leibniz may have had in mind with his quotient of infinitesimals.

physicists infinitesimals versus mathematicians differentials

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