1. ## Question involving Jacobian.

The equations:
$\displaystyle u^2 + v^2 + xy - x^2 + y^2 = 5$
$\displaystyle vxy - y^2 = 1$
Check the Jacobian condition at the point (x,y,u,v) = (1,1,0,2) and show that this allows x and y to be defined implicitly as functions of u and v. At this point, compute the Jacobian D(x,y)/D(u,v).

Ok, a few problems with this one.
1) I don't know what the Jacobian condition is.
2) I don't know how to show that x and y are functions of u and v.
3) I'm still iffy with Jacobians and I'm not sure how to find the Jacobian matrix.

So, any tips to help me through this number?

2. Originally Posted by bleepbloop
The equations:
$\displaystyle u^2 + v^2 + xy - x^2 + y^2 = 5$
$\displaystyle vxy - y^2 = 1$
Check the Jacobian condition at the point (x,y,u,v) = (1,1,0,2) and show that this allows x and y to be defined implicitly as functions of u and v. At this point, compute the Jacobian D(x,y)/D(u,v).

Ok, a few problems with this one.
1) I don't know what the Jacobian condition is.
2) I don't know how to show that x and y are functions of u and v.
3) I'm still iffy with Jacobians and I'm not sure how to find the Jacobian matrix.

So, any tips to help me through this number?
If $\displaystyle x = x(u,v)\;\;y=y(u,v)$ the Jacobian is defined as

$\displaystyle \frac{\partial (x,y)}{\partial (u,v)} = \begin{array}{| c c |}x_u & x_v\\y_u&y_v\end{array}$

where subscripts denote partial differentiation and $\displaystyle | \cdot|$ a determinant. Calculate your derivatives implicitly, substitute into the jacobian, then evaluate at your point.

3. Originally Posted by danny arrigo
If $\displaystyle x = x(u,v)\;\;y=y(u,v)$ the Jacobian is defined as

$\displaystyle \frac{\partial (x,y)}{\partial (u,v)} = \begin{array}{| c c |}x_u & x_v\\y_u&y_v\end{array}$

where subscripts denote partial differentiation and $\displaystyle | \cdot|$ a determinant. Calculate your derivatives implicitly, substitute into the jacobian, then evaluate at your point.
Thanks, but it tells me to find the Jacobian condition (what is this?) and to show how this implies that x and y are functions of u and v before going on to the Jacobian matrix.
How do I do these first two parts?