Necessary and sufficient conditions for differentiability of a function of several variable

OF A FUNCTION OF SEVERAL VARIABLES.

R.P. VENKATARAMAN,#1371, 13'TH MAIN ROAD,II STAGE, FIRST PHASE,

B.T.M. LAYOUT,BANGALORE 560 076.

arXiv:math.CA/0007011 v2 17 Jul 2000 ABSTRACT

The necessary and sufficient conditions for differentiability of a function of several realvariables stated and proved and its ramifications discussed.

Proposition: A scalar or vector valued function f of N real variables x1,x2,...xN isdifferentiable at (c1,c2,...cN) iff with respect to the point under consideration as

−2)

origin in spherical polar coordinates (rφ1,φ2,φ3,...φ(Nthe origin i.e.

,ϑ) it is differentiable at

exists, at r=0 i.e. it is independent of the angle variables.Proof.:

The limit of a function f(x1,x2,...xN)at (c1,c2,...cN) is independent of the choiceof the basis. Hence in spherical polar coordinates with(c1,c2,...cN) as origin if

∂fdf=dr

∂r

ltr→0fis independent of the angle variables(φ1,φ2,φ3,...φ(N−2),ϑ) then the limit

exists and the converse holds by the very definition of limit.

Derivative being a limit the result stated in the proposition follows. The differential ofa function of several variables is given by

z−z0=rexp(iϑ)

At r=0 it is easy to show that the partial derivatives of f with respect to the angle

variables are all zero since the arguments are all zero and hence the difference vanishes..Hence the proposition.

In the case of vector valued function the particular choice of the basis reduces thedivision to that by a scalar.

Cor.:1 A function f of complex variable z given by f(z)=u+iv is differentiableat z0 iff in polar coordinates defined by

the derivatives Pr and Qr of the real and imaginary parts of f exist at r

0

∂f

df=∇f.dr=dr

∂r

=0.

Cor.: 2 A function f of quaternion variable h=a+ib+jc+kd is differentiable at h iff inspherical polar coordinates with h as origin it is differentiable at the origin.

0

Discussion: The choice of spherical polar coordinates covers all possible paths and helpsus check easily if the limit exists and again find the limit easily whenever it exists. In the

case of differentiability of f(z)the proposition encompasses Cauchy-Riemann

conditions and leads to important conclusions. To check the continuity of the derivativeone has to choose a generic point as the origin. The problem of finding the maxima andminima of functions of several variables is drastically simplified to the problem ofsolving one equation in radial coordinate and checking the signature of the secondderivative in that coordinate. These are illustrated below with examples.

It has also equipped one to proceed with calculus on the division ring ofquaternions where multiplication is noncommutative.Examples:

Limit at the origin:1.

xy/[x2+y2]

exist as the angle variables are undefined at the origin.2. yexp(−1/polar coordinates it is

m

In the standard method it is shown that along y=mx, the limit is and hence

2

1+m

it does not exist. In polar coordinates it is ltr→0sinΘcosΘ and hence it does notx2)/[y2+exp(−2/x2)]

The limit is 0 along y=axnfor any n but is ½ along y=exp(−1/x2). In

ltr→0exp(−1/(r2cos2Θ)/(rsinΘ)

The above limit is infinity when Θ=0orΠ and is 0 otherwise and hence undefined.3. x2y2

/[x2+y2]

Differential of

In polar coordinates the limit is ltr→0(rsinΘcosΘ)2 and is found to be 0.Example 4. df

xy=1/2y/xdx+1/2x/ydy

does not exist at the origin as the partial derivatives do not exist at the origin.In polar coordinates dF

=sin(2Θ)/2dr+r/2cos(2Θ)/sin(2Θ)dΘ

does not exist as the angle is undefined at the origin.Example 5. Maxima and minima of f(x,y)

The procedure is to find the derivative of the function at a generic point P(a,b) byshifting the origin to this point using spherical polar coordinates and solve for theordered pair (a,b) by equating the derivative to zero

f(x,y)=xyexp(−xy)

df=[{y−xy2}exp(−xy)]dx+[{x−x2y}exp(−xy)]dy

In Cartesian coordinates one finds the solutions to be x=y=0andxy=1. From

the second derivatives it is easy to show that the second represents a maximum and the

first neither a maximum or a minimum as it does not satisfy coordinates F

ff>(f)2. In polar

xxyy

xy

=r2(sin2Θ)/2exp[−(r2/2)sin(2Θ)]

∂f/∂r=[rsin(2Θ)−r3(sin22Θ)/2]exp(−r2sin(2Θ)/2) ∂f/∂r=0 yields the same solutions as above: r=0andr2{sin(2Θ)/2}=1 .

The second derivative ∂2f/∂r2 does not exist at the origin and satisfies ∂2f/∂r2<0ifr2sinΘcosΘ=1

Thus the problem of finding the maxima and minima is reduced to solving just oneequation in r coordinate and checking the signature of the second derivative in thatcoordinate.

Example 6. In the examples below a generic point is chosen as the origin and f(z) issingle

_

I) Derivative of f(z)=z

Using z0 as the origin in polar coordinates, we find f'(z)=exp(−iθ)[exp(−iθ)]and hence it does not exist anywhere.II) Derivative of

2

f(z)=z

2

f(z)=z=(x−x)2+(y−y)2+(x2+y2)+2(x−x)x+2(y−y)yHence f'(z)=exp(−iϑ)[2r+2xcosϑ+2ysinϑ]

0

0

0

0

0

0

0

0

0

0

exists at (0,0) only.

f(z)=zn where n is an integer.

f'(z)=exp(−iΘ)[nr(n−1)cosnΘ+inr(n−1)sinnΘ] Since P(r,Θ) and Q(r,Θ) do not exist at r=0, if n< 1, f(z) is not

III) Derivative of

r

r

differentiable at r = 0 whenever n <1. It is easy to show as in the above two examples,by taking a generic point as the origin that the derivative in this case is a continuousfunction of z0 whenever n≥1.

IV)Proof of existence of higher derivatives for an analytic function from first principles: Setting t=rexp(iθ) we get

f'(z)

(z=z0)

=exp(−iϑ)[lt

(t→0)

The second derivative by definition is

d

f(t+z)]exp(iϑ)dt

0

f''(z)

(z=z0)

=exp(−iϑ)[lt

(∆t→0)

and since analyticity at z=z0 implies the continuity of f’(z)

f'(∆t,z)−f'(0,z)

]exp(iϑ)

∆t

0

0

[f''(z)]

(z=z0)

(δ→0)

where ε and δ are positive quantities. The above inequality shows f’’(z0) is finite. Thelimit is unique since r, and hence t, tend to zero only from the positive side. It is easy toshow that the second derivative is continuous since f’’(z0+∆z0) tends to f’’(z0) as ∆z0tends to zero. Hence, one can similarly show that the higher derivatives also exist.

This proof holds even for a function of a real variable, which is defined by one expressionin its entire domain, except that the values of θ are only zero and π. The same definitionof analyticity resulting in the existence of all higher derivatives holds also for a functionof several real variables and will be discussed in a subsequent article.

Thus continuity of f’(x) implies the existence of higher derivatives. The proof of

existence of Laurent’s series based on Fourier series given by Myskis1, and hence that ofTaylor series also hold for f(x). It is only while determining the radius of convergence ofthe series caution has to be exercised to take into account the singularities in the complexplane. The coefficients are given by the corresponding values for f(z). But in practice itcan be found directly using Taylor expansion of known functions. For example (1+x2)kwhere k is real, has a power series with radius of convergence 1 limited by the

singularities on the imaginary axis. It is easy to show that f(x) is analytic at x=±1 andhas a Taylor series expansion about x2 =1 whose radius of convergence is limited by thesingularities on the imaginary axis. That exp(-1/x2) does not have a power series isbecause 1/x2 is not analytic at the origin although exponential function is an entirefunction. Thus analyticity for a function of real variable, which is defined by one

expression in its domain, could be defined in the same way as for functions of complexvariable.

The choice of \"Spherical Polar Coordinates\" for this note would be equally appropriatethe result hinging as it does on this specific choice. Even in power series expansion offunctions of several variables this choice comes handy in addressing convergencequestions and also in the solution of first order differential equations :The equation

dy/dx=f(x,y)=−P(x,y)/Q(x,y)

can be solved easily in (r,Θ) using the method of separation of variables as long as P andQ are homogeneous in the two variables and of the same degree for

ε(δ)

]exp(iΘ)δdr/dϑ=

Prsinϑ−QrcosϑQsinϑ+Pcosϑ From the above it is easy to realise that singular points be rather defined as thosethrough which more than one curve passes violating Cauchy’s theorem. For, in thisbasis, for the equation governing a family of circles origin is not a singular point, thenumerator being zero and the denominator unity.

Taylor expansion in two real variables can be written as2

ij

f(x,y)=∑nxy+Ra(i+j)=0

ij

n

where Rn is the remainder term and if aij is defined as3

a=

ij

1(i+j)

[∂/∂xi∂yjf]i!j!

(x=0,y=0)

then Taylor series is defined as the limit of the above sum since convergence could beaddressed more easily in polar coordinates. For example 1)(1+xy)k2)

=1+k(xy)/1!+k(k−1)(xy)2/2!+......

2

is convergent for all (x,y)εR and kεR whenever in polar coordinates r is less than one.

exp(xy)=1+(xy)/1!+(xy)2/2!+.......+(xy)r/r!+....

is convergent for all (x,y)εR2. Power series for a few cases are given by Kaplan4.Extension to several variables is straight forward.

Conclusion :

Using spherical coordinates, necessary and sufficient conditions for the existence of thelimit of a scalar / vector valued function of several variables at a point and hence thosefor differentiability of the above functions at a point and hence those for differentiabilityof a function of complex variable and also quaternion variable have been stated andproved. The problem of finding the maxima/minima is drastically simplified to solvingjust one equation in radial coordinate and checking the nature of the second derivative inthat coordinate. The conditions for differentiability of f(z) encompasses Cauchy-Riemannconditions. It has been proved from first principles that analyticity of a function at apoint implies the existence of all higher derivatives and that the same definition is valideven for functions of real variable.. A few examples have been discussed to illustrate theconstructive method of finding the limit using the above choice of the basis.

No counter examples could be produced without violating the topological natureof limits. For further study functions of several complex variables, calculus onquaternions, solutions to differential equations about irregular singular points andramifications ,if any, on variational methods are being taken up.Reference:

1.Myskis A.D. (1975) Advanced Mathematics for Engineers –English Trans. VolosovV.M. and Volosov I.G., Mir Publishers.

2.Courant.R. & John.F (19) Introduction to Calculus and Analysis Volumes 1 and 2,Springer, New York.

3.Philip C.Curtis Jr. (1972) Multivariate Calculus with Linear Algebra, John Wiley andSons, New York.

4.Kaplan Advanced Calculus

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