Math fundamental: can linearity generate reciprocity?

In summary: Originally posted by Loren Booda In summary, we discuss the existence of a nontrivial function f(cx) satisfying (d/dx)f(cx)=(1/c)f(cx) where c is constant and "linearity" requires that c and x in the function argument preserve their product to the first power. It was concluded that such a function does exist, given that c is a nonzero constant, and can be represented as f(x)=C'exp(x/c). However, this function is not linear and the general solution for df/dx=(constant)f(x) is an exponential function. The statement (d/dx)f(c0x)=(1/c0)f(c0x) is not true for all c0, and only holds
  • #1
Loren Booda
3,125
4
Does a nontrivial function f(cx) exist such that

(d/dx)f(cx)=(1/c)f(cx)

and c is constant? "Linearity" here requires that c and x in the function argument preserve their product to the first power.
 
Mathematics news on Phys.org
  • #2
Originally posted by Loren Booda
Does a nontrivial function f(cx) exist such that

(d/dx)f(cx)=(1/c)f(cx)

and c is constant? "Linearity" here requires that c and x in the function argument preserve their product to the first power.


try f(cx) = exp(x/c)

c can be a constant
 
  • #3
Originally posted by Loren Booda
Does a nontrivial function f(cx) exist such that

(d/dx)f(cx)=(1/c)f(cx)

and c is constant? ...

Let's test it out. For any number c not equal to zero,
define f(cx) = exp(x/c)

Now check your equation

(d/dx)f(cx) = (d/dx)exp(x/c) = (1/c)exp(x/c) = (1/c)f(cx)

Therefore (d/dx)f(cx)=(1/c)f(cx)

so it works.

So the answer is YES a nontrivial function exists satisfying the equation with c a constant.
And any nonzero number c will do for the constant.
 
  • #4


Originally posted by marcus
try f(cx) = exp(x/c)

c can be a constant

I think if f(x) = exp(x) then
f(cx)= exp(cx) and df/dx = c * exp(cx)
 
  • #5


Loren, may I ask why you are looking for this function?
 
  • #6
1. c here is a contant and so has nothing to do with the "linearity" of f.


2. I think you are confusing things by always talking about "the function f(cx)". f(cx) does not represent a function, it represents a value of a function. f(x), f(y), f(cx) all refer to the same function, f.

Let y= cx. Then df/dx= df/dy dy/dx= c df/dx= c((1/c)f(cx))
= f(y) so you are requiring that c df/dy= f(y) or that
df/dy= (1/c)f(y). df/f= (1/c)dy so ln(f)= y/c+ C or

f(y)== C' ey/c.

That is, f(x)= C' ex/c as Marcus said.
 
  • #7
I forgot to say:

Of course, that function is not linear so the answer to the original question is "no".

In general, f satisfying df/dx= (constant) f(x) is exponential, not linear.
 
  • #8
Let y= cx. Then df/dx= df/dy*dy/dx= c*df/dx= c((1/c)f(cx))
= f(y) so you are requiring that c df/dy= f(y) or that
df/dy= (1/c)f(y). df/f= (1/c)dy so ln(f)= y/c+ C or

f(y)== C' ey/c.

That is, f(x)= C' ex/c as Marcus said.
Don't you mean df/dx=c*df/dy?
d[f(cx)]/dx=1/c[f(cx)]
if y=cx, dy=cdx
cd[f(cx)]/(cdx)=1/c[f(cx)]
cd[f(y)]/dy=1/c[f(y)]
df/dy=1/c2*f
df/f=dy/c2
lnf=y/c2+C1
f(y)=C2ey/c2
f(cx)=C2ex/c
 
Last edited:
  • #9
I am confused. Are you folks saying that (d/dx)f(c0x)=(1/c0)f(c0x) for all Aexp(c0x)=f(c0x)?

What about the falsehood (d/dx)Aexp(4x)=(1/4)Aexp(4x), for instance, where c0=4, or in general, where c0 is other than 1?
 
  • #10
No, we are saying that d[f(cx)]/dx=1/c[f(cx)] only if f(cx) is of the form
f(cx)=Cex/c

d[Cex/c]/dx=(1/c)Cex/c=1/c[f(cx)]

There is no way to obtain the reciprocal constant upon differentiation.
 
Last edited:

1. What is linearity in mathematics?

Linearity in mathematics refers to the property of a mathematical function or system that follows the principles of superposition and homogeneity. This means that the output of the function or system is directly proportional to the input, and the function or system remains unchanged when the input is scaled or added together.

2. How does linearity relate to reciprocity?

Linearity and reciprocity are closely related concepts in mathematics. Reciprocity refers to the mutual exchange or relationship between two quantities or systems. In the case of linearity, the property of superposition allows for the exchange of inputs and outputs, resulting in reciprocity.

3. Can linearity generate reciprocity in all mathematical systems?

No, linearity does not necessarily generate reciprocity in all mathematical systems. It depends on the specific properties and equations of the system. Some systems may have linear characteristics but do not exhibit reciprocity, while others may exhibit reciprocity without being entirely linear.

4. What are some real-life applications of linearity and reciprocity?

Linearity and reciprocity have numerous applications in various fields of science and engineering. In physics, the Laws of Motion and the principles of electricity and magnetism are based on linear and reciprocal relationships. In economics, supply and demand curves exhibit linear and reciprocal behavior. These concepts also play a crucial role in signal processing, control systems, and optimization problems.

5. How can linearity and reciprocity be used to solve problems in mathematics?

Linearity and reciprocity are powerful tools in mathematical problem-solving. By understanding the linear and reciprocal relationships between variables, equations and systems can be simplified and solved more efficiently. This can also help in predicting and analyzing the behavior of complex systems. Additionally, the properties of linearity and reciprocity can be used to prove theorems and solve equations in various branches of mathematics.

Similar threads

  • General Math
Replies
7
Views
831
Replies
6
Views
2K
  • General Math
Replies
16
Views
3K
  • General Math
Replies
0
Views
586
  • General Math
Replies
13
Views
1K
  • Precalculus Mathematics Homework Help
Replies
5
Views
924
  • General Math
Replies
2
Views
1K
Replies
1
Views
1K
Replies
3
Views
1K
Replies
3
Views
740
Back
Top