Change of variables; why do we take the absolute value?

In summary, when transforming an integral to new coordinates, the "volume" element is multiplied by the absolute value of the Jacobian determinant. However, in the one-dimensional case, where "change of variables" is just "substitution," the absolute value of the derivative is not necessary. This is because in the one-dimensional case, the integral borders will change signs accordingly. In the multi-dimensional case, the integral borders cannot be used to keep track of the sign. It is important to note that the change of variables formula only tells you the region you are integrating over, not the bounds of the integral. This means that when writing it as an iterated integral, each integral should go from the lower bound to the upper bound. This is
  • #1
Hiero
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In transforming an integral to new coordinates, we multiply the “volume” element by the absolute value of the Jacobian determinant.

But in the one dimensional case (where “change of variables” is just “substitution”) we do not take the absolute value of the derivative, we just take the derivative, be it positive or negative.

Why is the single-variable method different from the multi-variable method (in that it lacks the absolute value)?
 
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  • #2
In the one-dimensional case your integral borders will also change signs accordingly, so you don't need the absolute value. In the multi-dimensional case you can't use the integral borders to keep track of the sign.
 
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  • #3
mfb said:
In the multi-dimensional case you can't use the integral borders to keep track of the sign.
Can you elaborate? We still change the boundaries according to the transformation (or it’s inverse) right? Doesn’t it depend on the choice of parameters? A simple example, (x, y)=(-u, v) (I mean a reflection across the y axis,)
$$1 = \int_0^1\int_0^1dxdy = \int_0^1\int_0^{-1}|J|dudv=-1$$
Why is that wrong to do? x = 1 corresponds to u = -1
 
  • #4
I think in this case you shouldn't change the integration borders, or take the absolute value of the overall integral.
 
  • #5
Is it just an unspoken part of the rule that we must make the integral over the region so that every inner integral always ‘runs upwards’? I.e. if we have an integral with lower-bound a and upper-bound b which are functions like b = b(x, y, ..., z) then we must make sure that b > a over the whole domain (x, y, ... z) which we might be integrating over?

Then I can understand it working with the absolute value |df/dx| in the 1d case, as well as how the previous example should go from 01∫∫-10dudv
 
  • #6
The actual statement of the change of variables formula is that [itex]\int_S f dA=\int_T f\circ\phi |J| dA[/itex] where [itex]\phi[/itex] is diffeomorphism from [itex]T[/itex] to [itex]S[/itex] (In your example, [itex]\phi(u,v)=(-x,y)[/itex]). Note that this formula doesn't give you iterated integrals with bounds; it only tells you the region that you're integrating over. You're right than when you write it as an iterated integral, each integral should go from the lower bound to the upper bound- this is because, for example, [itex]\iint_{[0,1]\times [-1,0]}dA[/itex] unambiguously equals [itex]\int_0^1\int_{-1}^0 du dv[/itex]
 
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  • #7
Infrared said:
when you write it as an iterated integral, each integral should go from the lower bound to the upper bound-
Thanks. I was not distinguishing the idea of a “multiple-integral” from an “iterated-integral,” which caused me confusion.

Also, I should have read “Fubini’s theorem” more carefully, because it does explicitly state the upper bounds are larger than the lower bounds (so it’s not an unspoken rule, it’s just spoken in a different theorem).
 

Related to Change of variables; why do we take the absolute value?

1. Why do we need to use a change of variables in mathematics?

The use of a change of variables allows us to simplify equations and make them easier to solve. It also allows us to study different aspects of a problem by transforming it into a new form.

2. What is the purpose of taking the absolute value when using a change of variables?

Taking the absolute value ensures that the new variable remains positive, which is necessary for certain mathematical operations and interpretations.

3. Can we use any variable for a change of variables?

Yes, we can use any variable as long as it helps us simplify the problem and leads to a more straightforward solution.

4. How do we know when to use a change of variables in a problem?

We should use a change of variables when the original problem is too complex or difficult to solve directly. It is also useful when trying to find different ways to look at the problem or when trying to prove a theorem.

5. What happens to the original problem after a change of variables?

The original problem is transformed into a new form, making it easier to solve. However, the solution to the new problem can be related back to the solution of the original problem through the change of variables.

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