The product of two functions

[Viona]

Hello

if we are given that for any functions g and f where:
g(x)*f(x)=g(y)*f(y)

is this requires that g(x)*f(x)=g(y)*f(y)=constant?
is this a theory? does it have a proof?

Thanks

 

[Dr.Peterson]

if we are given that for any functions g and f where:
g(x)*f(x)=g(y)*f(y)

is this requires that g(x)*f(x)=g(y)*f(y)=constant?
is this a theory? does it have a proof?

I am not sure of the wording of your question. It is not true that for any two functions, g(x)*f(x)=g(y)*f(y) for all x and y (or even for any specific x and y).

My best guess is that you meant to say something like this:

If for given functions f and g, g(x)*f(x)=g(y)*f(y) for all x and y, can we conclude that g(x)*f(x) is constant?
​

If this is related to a specific problem, please state it word for word, so we can be sure what you are asking. The more you can say about the context of your question (why you are asking, any constraints, etc.), the better we can answer.

 

[Viona]

I am not sure of the wording of your question. It is not true that for any two functions, g(x)*f(x)=g(y)*f(y) for all x and y (or even for any specific x and y).

My best guess is that you meant to say something like this:

If for given functions f and g, g(x)*f(x)=g(y)*f(y) for all x and y, can we conclude that g(x)*f(x) is constant?
​

If this is related to a specific problem, please state it word for word, so we can be sure what you are asking. The more you can say about the context of your question (why you are asking, any constraints, etc.), the better we can answer.

I was reading in a book of Molecular Physics. The topic is Kinetic Theory of Ideal Gases(Distribution of Molecules by Velocity Components). Please see the attached picture. f is the distribution function of molecules by velocity components.

 

[Dr.Peterson]

I was reading in a book of Molecular Physics. The topic is Kinetic Theory of Ideal Gases(Distribution of Molecules by Velocity Components). Please see the attached picture. f is the distribution function of molecules by velocity components.
View attachment 10558

Is the first equation correct, with positive exponent on one side and negative on the other, or is there a typo?

The right side is constant (i.e. independent of v_z); assuming this equation is supposed to be true for all v_z, this implies that the left side is equal to a constant, which you can call A. Can you finish?

 

[Viona]

Is the first equation correct, with positive exponent on one side and negative on the other, or is there a typo?



The right side is constant (i.e. independent of v_z); assuming this equation is supposed to be true for all v_z, this implies that the left side is equal to a constant, which you can call A. Can you finish?

Yes it is a typo. sorry.

Also the left side is constant (i.e. independent of v_zo), this implies that the left side equals to the same constant A.
Now I understood what they mean. Thanks a lot.