Systems of equations using partial derivatives

[sandy.bridge]

Homework Statement

Consider the system of equations
$$x^2y+za+b^2=1$$
$$y^3z+x-ab=0$$
$$xb+ya+xyz=-1$$

1. Can the system be solved for $$x, y, z$$ as functions of $$a$$ and $$b$$ near the point $$(x, y, z, a, b)=(-1, 1, 1, 0, 0)$$?

2. Find $$\frac{\partial x}{\partial a}$$ where $$x=x(a, b)$$

The Attempt at a Solution

For part 1, do I merely plug in $$(x, y, z)=(-1, 1, 1)$$ and then solve the system of linear equations with the variables a and be remaining in the function?

Part 2 really confuses me. My textbook has taught us one notation, and then it states a question in this format. For example, often it will say something such as $$(\frac{\partial z}{\partial x})_w$$ to notify when something is independent or dependent. Can someone clarify what the question is stating?

 

[dimpledur]

Not sure about part 1. I believe part two is telling you to take the partial derivative of the system of linear equations with respect to a, all the while regarding both a and b as independent variables...
Someone can clarify this.

 

[sandy.bridge]

That can't be true...

 

[dimpledur]

Okay... I just looked at my notes, and part 1 can be solved via Jacobian matrix. Take the determinant of the matrix, and if this value is nonzero, then that assures us these can be solved with respect to x, y, z.

 

[sandy.bridge]

Okay, thanks.

For part 2, x=x(a, b) is stating that x is the dependent variable, and a and b are both independent variables. Can anyone clarify where the y and z variables stand in this instance? How do I tell if they are dependent, or independent variables.

Thanks

 

[dimpledur]

x, y, z are dependent, and a and b are independent.

 

[sandy.bridge]

okay, i figured out both part 1 and 2. Does anyone know if I should be plugging in the values from part 1 into x, y, z, a, b to get a numerical answer, or should I leave it in terms of x, y, z, a, b?