Proportionality with more than one variable?

[kochibacha]

if x is direct, indirect or exponentially propotional to A and as well as B

can we write x=kAB ? if we write the equation seperately, we have x=k1A, x=k2B when combined, x2=(k1k2)1/2 (AB)2 then x=k3(AB)1/2

to see the real complicate example

EX.1 trypsinogen is converted to trypsin in the body where trypsin itself catalyzes its own reaction

let f(t) = amount of trypsin at time t
let F(t) = amount of trypsinogen at time t

write differential equation satisfied by f(t)

in short, f(t) is direct proportional to product itself f(t) and the substrate F(t)

should i write f'(t)=k f(t)F(t) , =k ( f(t)+F(t) ) , = k (f(t)F(t))1/2 or something else?

 

[jbriggs444]

if x is direct, indirect or exponentially propotional to A and as well as B

If x is directly proportional to A then there is a constant k such that x = kA.
If x is inversely proportional to A then there is a constant k such that x = k/A.

I have never encountered the terms "indirect proportionality" or a "exponential proportionality". Fortunately, those terms are irrelevant to the questions below.

can we write x=kAB ? if we write the equation seperately, we have x=k1A, x=k2B when combined, x2=(k1k2)1/2 (AB)2 then x=k3(AB)1/2

Yes, we can write x = kAB.

[Editted to eliminate my first erroneous explanation]

It is tempting to multiply the two equations together to get x² = k₁k₂AB

The problem is that the x=k₁A is true only as long as one holds B constant. The value of k₁ includes that constant value of B. Similarly, x=k₂B is true only as long as one holds A constant. The value of k₂ includes that constant value of A.

EX.1 trypsinogen is converted to trypsin in the body where trypsin itself catalyzes its own reaction

let f(t) = amount of trypsin at time t
let F(t) = amount of trypsinogen at time t

write differential equation satisfied by f(t)

in short, f(t) is direct proportional to product itself f(t) and the substrate F(t)

It is the reaction rate that is proportional to the product of f(t) and F(t). So rather than f(t) being proportional to itself (a trivial tautology), it is f'(t) that is proportional to f(t).

So the short form would be "f'(t) is directly proportional to the product of f(t) and F(t)"

 

[kochibacha]

sorry for ambiguous writing it must be X=K(AB)^1/2 and f'(t) is direct proportional to product itself f(t) and the substrate F(t)

However, your answer still hasn't answered my questions.

if x is directly proportional to A and as well as B. How can we express x in terms of equations?

if you answered x=constant*A*B could you explain why not x=constant*(A+B)
and what about x=constant₁*A , x=constant₂*B

when combined, x²=constant₁₊₂*AB

x=+K(AB)^1/2 and x=-K(AB)^1/2 but we ignore the minus one so x=K(AB)^1/2

 

[jbriggs444]

if you answered x=constant*A*B could you explain why not x=constant*(A+B)

Suppose for a moment that the above formula held: x = k(A+B) for some constant k.
But we also know that x=k'A for some constant k'.

Take A=1, B=2. Then x=3k by the one equation and x=k' by the second. So k'=3k.
Take A=2, B=1. Then x=3k by the one equation and x=2k' by the second. So 2k'=3k.

Clearly, the only way this can hold is if both k and k' are equal to zero. So x=constant*(A+B) cannot be right except in the degenerate case where x is always zero.

and what about x=constant₁*A , x=constant₂*B

when combined, x²=constant₁₊₂*AB

As I wrote, because that constant₁ is not a constant. It is a function of B. Similarly, constant₂ is not a constant. It is function of A.

What function of A can work for constant₁? constant₁ = k₁B can work.

What function of B can work for constant₂? constant₂ = k₂A can work.

What do you get when you multiply the two equations together now?

 

[CellCoree]

Proportionality with more than one variable can be seen in various scientific phenomena. In the case of x being proportional to both A and B, we can write the equation x=kAB. This can also be written as x=k1A and x=k2B. When combined, we can rewrite it as x2=(k1k2)1/2 (AB)2, which simplifies to x=k3(AB)1/2. This shows that x is directly proportional to the square root of the product of A and B.

In a more complicated example, let's consider the conversion of trypsinogen to trypsin in the body. We can define f(t) as the amount of trypsin at time t and F(t) as the amount of trypsinogen at time t. The differential equation satisfied by f(t) can be written as f'(t)=k f(t)F(t). This shows that the amount of trypsin is directly proportional to the product of itself and the substrate trypsinogen.

In short, when dealing with proportionality with more than one variable, we can write the equation as x=kAB or x=k1A and x=k2B. When combined, it can be simplified to x=k3(AB)1/2. In the example of trypsinogen conversion, the differential equation can be written as f'(t)=k f(t)F(t). This shows that the relationship between multiple variables can be expressed through proportionality and understanding this relationship is important in various scientific fields.