If u = xyz, v = x2 + y2 + z2, w = x + y + z, find J = racd(u,v,w)d(x,y,z).

Here are the solutions to the questions:

QUESTION ONE

(a) (i) If $u = xyz$, $v = x^2 + y^2 + z^2$, $w = x + y + z$, find $J = \frac{\partial(u,v,w)}{\partial(x,y,z)}$.

Step 1: Calculate the partial derivatives of $u, v, w$ with respect to $x, y, z$.

$$\frac{\partial u}{\partial x} = yz, \quad \frac{\partial u}{\partial y} = xz, \quad \frac{\partial u}{\partial z} = xy$$ $$$$