Easy matrix integration question

[Rickster26ter]

Homework Statement

This is just the triple integral of an easy matrix problem. I just have no ideas what they got by the time they got to the integral of x.

Homework Equations

integral[/B]

The Attempt at a Solution

Somebody please prove me wrong. I got a matrix of constants by the time I got to x, not this

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[Orodruin]

Please show us what the actual problem is and your own efforts to solve it.

 

[Rickster26ter]

 

[Rickster26ter]

View attachment 217149

Notice, the main problem is that after I integrate, a z is mutliplied by everything, and then turned into 0.

 

[Orodruin]

Clearly the integral is zero. The integration domain has measure zero. I suspect what was intended was to integrate some delta functions over a non-empty domain.

It would help if you stated the problem. Always state the full problem exactly as given.

 

[FactChecker]

This is a little unusual. I think that the answer is the zero matrix. The reason is that even the elements that do not include y or z would integrate to 0 when the integral ends are both 0. In fact any time both ends are the same. Suppose the antiderivative of function f is F. Then ∫_a ^af(y) dy = F(y)|_y=a - F(y)|_y=a = F(a)-F(a) = 0

 

[Orodruin]

This is a little unusual. I think that the answer is the zero matrix. The reason is that even the elements that do not include y or z would integrate to 0 when the integral ends are both 0. In fact any time both ends are the same. Suppose the antiderivative of function f is F. Then ∫_a ^af(y) dy = F(y)|_y=a - F(y)|_y=a = F(a)-F(a) = 0

This is clearly the case. I suspect the integral has been malconstructed. It seems to be the computation of the moment of inertia tensor for a slender rod, except the density function should be a delta function in y and z, not a finite one. As it stands, the dimensions are even wrong.

 

[Rickster26ter]

Yea, I realized what I did wrong with trying to at least get constants off, it should be 0. But how could this be? Let me give everyone a zoom out on how this should apply to physics a little better.

 

[Rickster26ter]

Can you explain or show to me what the integral should look like then?

 

[Rickster26ter]

I'm an electrical Engineer, and have only dealt with delta functions in wave properties, not tensors, so can you explain further?

 

[Orodruin]

I suggest that you think about what dm is in terms of the coordinates. In general, it is given by $dm = \rho \, dV$ and you should integrate over all of space. (There are several hints in my previous posts.)

For the future: You should have given us the entire problem in your first post. It would have significantly simplified the process. As it was I had to guess what you were actually up to.

 

[Rickster26ter]

Fair enough. I'm still a bit confused as to what is "malconstructed." I think I understand why you are saying we need delta functions, but I still don't get why how the integral bounds should be constructed, if not 0 to 0, and how that gives us the (2,2) and (3,3) x^2 values..

 

[Rickster26ter]

If you want me to show my work, somebody who knows less, how much more do I need you to show work of what you are talking about.

 

[Orodruin]

I'm still a bit confused as to what is "malconstructed."

For example, if you would try to compute the mass in a similar way, you would get zero.

One way around using delta functions is to skip the y and z integrations altogether and write $dm = \rho_\ell dx$, where $\rho_\ell$ is the linear density (mass per length). The y and z coordinates are always zero.

 

[vela]

Fair enough. I'm still a bit confused as to what is "malconstructed." I think I understand why you are saying we need delta functions, but I still don't get why how the integral bounds should be constructed, if not 0 to 0, and how that gives us the (2,2) and (3,3) x^2 values..

You could integrate from $y=-\infty$ to $y=\infty$. It actually doesn't really matter what interval you use as long as it contains $y=0$ because the delta function is zero except when $y=0$, where the rod lies. Same with $z$.

The author of the book was extraordinarily sloppy. There are often cases where you write mathematics down which isn't rigorously correct but which gets the job done, but the integral you're dealing with is just plain wrong. It's no surprise it confused you.