Explicit Equations for Implicit Set at (0,0): First Partial Derivatives

[minderbinder]

Homework Statement

Show that the set defined by the equations
x + y + z + w = sin(xyzw)

<br /> x - y + z + w^2 = cos(xyzw) - 1
can be described explicitly by equation of the form (z, w) = f(x, y) near the point (0,0,0,0); find the first partial derivatives of f(x,y) at the point (0,0)

Homework Equations

The above bolded part is the part I'm unsure about...

The Attempt at a Solution

I did:
<br /> G = x + y + z + w - sin(xyzw)

<br /> H = x - y + z + w^2 - cos(xyzw) + 1

<br /> \frac{\partial (G, H)}{\partial (x, y)} + \frac{\partial (G, H)}{\partial (z, w)} \frac{\partial f}{\partial x } = 0<br />

Then I solved for \frac{\partial f}{\partial x}?

 

[Fenn]

I think the note of "near the point (0,0,0,0)" is a clue that you can use the small angle approximation of

<br /> \sin\theta = \theta,\quad<br /> <br /> \cos\theta = 1,\quad<br /> <br /> \texttt{when }\theta \texttt{ is small.}<br />

I'm not fully understanding what the question is asking, but I'm interpreting it as saying you need to find

<br /> z = f_1(x,y)<br />

<br /> w = f_2(x,y)<br />

From there, I would calculate
<br /> \left.\frac{\partial z}{\partial x}\right|_{(0,0)} = \frac{\partial f_1(0,0)}{\partial x}<br />
<br /> \left.\frac{\partial z}{\partial y}\right|_{(0,0)} = \frac{\partial f_1(0,0)}{\partial y}<br />
<br /> \left.\frac{\partial w}{\partial x}\right|_{(0,0)} = \frac{\partial f_2(0,0)}{\partial x}<br />
<br /> \left.\frac{\partial w}{\partial y}\right|_{(0,0)} = \frac{\partial f_2(0,0)}{\partial y}<br />

As I said, though, I may not be properly understanding the question.