Change of Variables in Tripple Integrals

[unscientific]

Homework Statement

In double integrals, the change of variables is fairly easy to understand. With u = constant and v = constant, along line KL v = constant so dv = 0. Therefore the only contributing variable to ∂x and ∂y is ∂v.

The Attempt at a Solution

However, in tripple integrals, you're simply adding one more w = constant that gives the 3rd dimension (the height to the initial flat-surfaced parallelogram before).

So as you move along a line PQ (which is formed by say u = constant), then won't BOTH v and w contribute to ∂x, ∂y and ∂z since only u is constant?

 

[Millennial]

Are you aware of the Jacobian determinant?

 

[Muphrid]

In 3D, the object defined by a constant coordinate is a surface, not a line. z = 0 defines a plane, for instance, doesn't it?

They're saying that along PQ two coordinates are constant because that's where two surfaces of (different) constant coordinates intersect.

In short, for 3D:

No coordinates constant: a volume
1 coordinate constant: a plane
2 coordinates constant: a line
All 3 coordinates constant: a point

 

[unscientific]

In 3D, the object defined by a constant coordinate is a surface, not a line. z = 0 defines a plane, for instance, doesn't it?

They're saying that along PQ two coordinates are constant because that's where two surfaces of (different) constant coordinates intersect.

In short, for 3D:

No coordinates constant: a volume
1 coordinate constant: a plane
2 coordinates constant: a line
All 3 coordinates constant: a point

Hmm I think I got it! For example, u = constant family of curves define the x-y plane, while v = constant define the y-z plane etc... the line is at intersection of the 2 planes where they cut one another.

Therefore, since the line lies on both plane x-y and y-z, v and u must be constant!

 

[unscientific]

Am I right to say that in the new coordinate surfaces u, v and w they need not necessarily be at right angles to one another as in x, y and z?

 

[unscientific]

In the picture attached there is a parallelepiped. My question is, what happens if plane ABCD is not vertical, like it slants into the page then the corresponding shape will not be of a parallelpiped? As in it slants into the page, making an angle with AC.

 

[Muphrid]

Am I right to say that in the new coordinate surfaces u, v and w they need not necessarily be at right angles to one another as in x, y and z?

That's correct. The coordinate system need not be orthogonal.



The parallelipiped is still a parallelpiped as long as the corresponding surface opposite ABCD slants the same way.