CONFUSED ABOUT WHEN TO LET x = Another x WEird Question Help

[PhysicsHelp12]

Hi, I'm going into 2nd year as a math major ...In the last week I've been thinking consciously about how if an algebraic property is written in terms of x ...then it applies to x^2 ,x-2 x^3, etc .

Like:

sin(2x)=2sin(x)cos(x)

Implies sin(2x^2)=2sin(x^2)cos(x^2)

I know this without thinking about it ...but

somehow letting x=x^2 confuses me ...I know theyre differnent xs technically...but I don't

want to have to think about that everytime I do a question like this ...it slows me down.


so When am I supposed to know that if an "rule" is given in terms of just x or n, etc

that it applies to all functions of that variable ...when am I supposed to consider this


if that makes sense

 

[HallsofIvy]

You're in your second year as a math major? I would have thought that in elementary algebra you would have learned that you can always call something by another name, as long as you are consistent. And that's all you are talking about.

You can certainly replace x by, say y or b or 2y or 2x or anything you like as long as you replace it everywhere in your formula.

If you have sin(2x)=2sin(x)cos(x) then you also have sin(2a)= 2sin(a)cos(a) or sin(2b)= 2sin(b)cos(b) or sin(2x²)= 2sin(x²)cos(x²). You could NOT say sin(2x)= 2sin(y)cos(y) or sin(2x^2)= 2sin(x)cos(x) because you haven't been consistent: you haven't replace every x with the new "label".

There is no "rule" except that variables and expressions are simply "labels" and what label you use is completely arbitrary- as long as you are consistent.

 

[atyy]

Well, log(xy)=log(x)log(y) means log(10.20)=log(10)log(20) or log(4.5)=log(4)log(5), but it does not mean log(-1.4)=log(-1)log(4), because log(x) is not defined for negative numbers.

So to be a bit more precise, sometimes people will say

log(xy)=log(x)log(y) for all x,y that are positive real numbers.

This makes it clear that you can replace x with anything that is also a positive real number.

There's a confusing bit in integration of multiple variables where sometimes you have to change variable names (ie. it is not optional), and the other thing is the dxdy under the integral is not quite the same as dx,dy outside the integral, because the former is short hand for a cross product that gives an area.

 

[Redbelly98]

somehow letting x=x^2 confuses me ...

They're not actually being set equal. It's a matter of substituting or replacing one term with another term.

 

[schroder]

Hi, I'm going into 2nd year as a math major ...In the last week I've been thinking consciously about how if an algebraic property is written in terms of x ...then it applies to x^2 ,x-2 x^3, etc .

Like:

sin(2x)=2sin(x)cos(x)

Implies sin(2x^2)=2sin(x^2)cos(x^2)

I know this without thinking about it ...but

somehow letting x=x^2 confuses me ...I know theyre differnent xs technically...but I don't

want to have to think about that everytime I do a question like this ...it slows me down.


so When am I supposed to know that if an "rule" is given in terms of just x or n, etc

that it applies to all functions of that variable ...when am I supposed to consider this


if that makes sense

If I understand your question correctly, what you are asking about is the difference between an Identity and an Equation. The first example you gave : sin(2x^2)=2sin(x^2)cos(x^2), is an Identity, which means it holds true for ALL values of x. The second example you gave : x=x^2 is an equation which only holds true for specific values of x. x = x^2 can only be true fo x =1, or x = 0 and for no other real value of x so it is not an identity. As you gain more experience with math you will be able to recognize the difference.

 

[NoMoreExams]

Well, log(xy)=log(x)log(y) means log(10.20)=log(10)log(20) or log(4.5)=log(4)log(5), but it does not mean log(-1.4)=log(-1)log(4), because log(x) is not defined for negative numbers.

So to be a bit more precise, sometimes people will say

log(xy)=log(x)log(y) for all x,y that are positive real numbers.

This makes it clear that you can replace x with anything that is also a positive real number.

There's a confusing bit in integration of multiple variables where sometimes you have to change variable names (ie. it is not optional), and the other thing is the dxdy under the integral is not quite the same as dx,dy outside the integral, because the former is short hand for a cross product that gives an area.

I'm not sure which people will say "log(xy)=log(x)log(y) for all x,y that are positive real numbers." since log(xy) = log(x) + log(y)

You are confused as people mentioned with substitution. The identity is

sin(2*anything) = 2*sin(anything)*cos(anything)

Now anything can be... well anything, x, 2x, 472478278247824782x, etc.

 

[HallsofIvy]

Well, log(xy)=log(x)log(y) means log(10.20)=log(10)log(20) or log(4.5)=log(4)log(5), but it does not mean log(-1.4)=log(-1)log(4), because log(x) is not defined for negative numbers.

So to be a bit more precise, sometimes people will say

log(xy)=log(x)log(y) for all x,y that are positive real numbers.

This makes it clear that you can replace x with anything that is also a positive real number.

There's a confusing bit in integration of multiple variables where sometimes you have to change variable names (ie. it is not optional), and the other thing is the dxdy under the integral is not quite the same as dx,dy outside the integral, because the former is short hand for a cross product that gives an area.

Please, please, please! The question was "when can you replace x by something else". my response was that you can replace x by anything as long as you are consistent (of course, you have to consider the domain of the functions). Your response, that "you can replace x with anything that is also a positive real number" is valid only for this particular example because the domain of log x is "all positive real numbers".

But my real reason for complaining is "sometimes people will say log(xy)=log(x)log(y) for all x,y that are positive real numbers". Not many people, I hope, will say that because it is not true! What is true is that log(xy)= log(x)+ log(y) whenever x and y are positive real numbers.

 

[Xs1t0ry]

Please, please, please! The question was "when can you replace x by something else". my response was that you can replace x by anything as long as you are consistent (of course, you have to consider the domain of the functions). Your response, that "you can replace x with anything that is also a positive real number" is valid only for this particular example because the domain of log x is "all positive real numbers".

But my real reason for complaining is "sometimes people will say log(xy)=log(x)log(y) for all x,y that are positive real numbers". Not many people, I hope, will say that because it is not true! What is true is that log(xy)= log(x)+ log(y) whenever x and y are positive real numbers.

Amen. I was scared for a minute.

 

[atyy]

1. Here is an example where you can change variable on one side of the equation only. Let DI(f(x),x) mean definite integral of f(x) with respect to x.

Then this equation in trivially true, since it's the same on both sides:

DI(f(x),x)=DI(f(x),x)

But it will still be true if we change only the right side:

DI(f(x),x)=DI(f(u),u)=DI(f(z),z) etc.

This is because x is being "integrated out" in a definite integral. You will hear people say that the variable in a definite integral is a "dummy variable" where you can change only one side of the equation, as opposed to the "free variable" in your original example. But you still have to be careful. Suppose:

f(x)=DI(g(x,y),y)

You can replace y by anything *except* x, ie.

f(x)=DI(g(x,x),x) is wrong!

But

f(x)=DI(g(x,u),u) is right.

This is also correct:

f(u)=DI(g(u,x),x).

2. Another thing to be careful about is that in mathematics:

P(x,y)=x²+y² means P(r,q)=r²+q²

But in physics, people often write

P(x,y)=x²+y²=P(r,q)=r²cos²(q)+r²sin²(q)=r²

where they have switched from cartesian to polar coordinates.

Of course, the mathematicians are more correct, and the physicists are being lazy. But it is very common notation in physics, so you should get used to that also. While it is more efficient to be able to do maths without thinking, it is often useful to think of what your equation means, by associating it with some physical situation, for example. So it is good that you stop and think about what you are doing. That way you will see why in a double integral, the dxdy are not ordinary products, but are actually cross products, and hence you need a Jacobian when you change variables (don't worry about this last statement now if you don't understand it, it's just another situation where you should not change variables blindly).