M371 - Alexiades
          Problem Set 4:   Nonlinear Systems

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For each (nonlinear) system below:
1. Write it in vector form:  F(x,y) = 0   to identify the components of the vector-valued function F(x,y).
2. Sketch/plot the curves to see if there are any solutions (intersections).
3. Write down the Jacobian J(x,y) matrix, and the (linear) system for Newton iteration.
4. Carry out one Newton iteration, by hand, starting with some initial guess (x₀ , y₀).

A.  x + y = 3 ,   x² + y² = 16

B.  y = x² ,   x² + (y − 2)² = 4

C.  x² − x y + y² = 21 ,   x² + 2 x y − 8 y² = 0