How to simplify this equation with change of variables,

Question

I have the equation, after completing the square:

$$(x+\frac{y}{2})^2 + \frac {3y^2}{4} + z^2 = 1$$

How can I further simplify this equation? I need to find the volume inside of this surface.

Changing to u,v,w coordinates would probably be the next move, but if I let $$u = x+\frac{y}{2}$$

what should v, w be? The second term of the equation depends on y, and hence would also be related to the u-variable, which seems problematic - u and v would then be related variables instead of independent variables.

Thanks,

Answer 1

All that is required by the change of variables is that it is 1-1 and onto. Letting $v=y$ and $w=z$ will work just fine. In matrix form, this is

$$\begin{bmatrix}u\\v\\w\end{bmatrix}=\begin{bmatrix}1 & 1/2 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}.$$

Since the determinant is nonzero, the change of variables is 1-1 and onto.

Answer 2

It's not a real answer, but may serve for you, since you just want to calculate the volume. Just doing the transformation you suggested, on can write:

$u^2 + \frac{3}{4}y^2 + z^2 \leq 1$

Whose Jacobian of this transformation is 1.

So, you can calculate the volume of this ellipsoid, which is the same as the one you're looking for:

$\frac{u^2}{1^2} + \frac{y^2}{(2/\sqrt{3})^2} + \frac{z^2}{1^2} \leq 1$

Then $V = \frac{4}{3}\pi (1)(2/\sqrt{3})(1)=\frac{8 \pi}{3/\sqrt{3}}$.