Triple integral - volume
I need to find the volume of this integral.
I'm pretty sure I did the second part fine, but I'm really not sure about the jacobian, can I do that?
And another thing, I know I'm asking a lot of question these past few days, but it's only because I have a big test on Sunday (Headbang)
I assume the volume you are trying to find is the set (where ).
The Jacobian theorem says,
Here is a differenciable vector function.
And is the determinant of the Frechet' derivative i.e. the Jacobian.
Of course, is the set
It is conventient to have be such a function so that .
If we let then .
This means, the function (thinking of vectors in as coloums).
Now, this is convenient because
The volume of is given by,
Thus the volume is because the volume of a sphere is
Originally Posted by ThePerfectHacker
So I was suppose to take , right?
I am not sure what you mean by this.
Originally Posted by asi123
You probably mean the reciprocal of the determinant you computed.
Maybe you did not understand what I did - I know I do Jacobians my way so maybe it was confusing.
I mean that I calculated that Jacobian with the first equation (in the pic) instead using the second one.
I switched to u=u(x,y,z), v=(x,y,z) and R=(x,y,z) and as I was saying used the first one instead of the second.
And my final answer would have also been if I would have taken Jacobian so I think that was my mistake.