Question

10. (Sections 11.1 - 11.3, 7.5, 9.3) (a) State the Implicit Function Theorem for an equation in the three variables x, y and z. (2) (b) Consider the equation xyz = 4x 2 + y 2 − z 2 . Use the Implicit Function Theorem to show that the given equation has a smooth unique local solution of the form z = g(x, y) about the point (2, 0, 4). Then find the local linearization of g about the point (2, 0). Hints: • Use the method of Example 11.2.6, but take into account that you are dealing with an equation in three variables here and that g in this case is a function of two variables. • Before you apply the Implicit Function Theorem, you should show that all the necessary conditions are satisfied. • Study Remark 11.3.3(1) and Remark 9.3.6(2).

Answer 1

1.(a) The implicit function theorem addresses a question with two versions;

• The analytic version implying a question about finding a solution of a system of non linear equations
• The geometric version about the geometric structure of certain sets.

Assuming $s$ is an open subset of $R^{n+k}$ and that $F:S\to R^k$ is a function of class $C^1$ . Assume also that $(a,b)$ is a point in that

$F(a,b)=0$ and det $D_yF(a,b)\not =0$

Consider a continuously differentiable function $F(x,y,z)=c$ such that

$dF\over dz$ $(x,y,z)\not=0$ then $F$ is $\alpha(x,y,z)$ such that $(x,y)$ is sufficiently close to $(x,y)$ then $F(x,y,z)=C$

10.(b) Given , $xyz=4x^2+y^2-z^2....(i)$

Let $F=4x^2+y^2-z^2-xyz$

$F=4x^2+y^2-z^2-xyz$

$\nabla F=[8x-yz,2y-xz,-2z-xy]$

$\nabla F(2,0,4)=[8\times 2-0\times 4,2\times 0-2\times 4, -2\times 4-2\times 0]$

$=[16, -8,-8]$

Equation of linearization

$z=z_o+F_x(x-xy)+F_y(y-y_1)....(ii)$

$z_o(x=2,y=0)$ then $z=z_o$

From equation $(ii)$

$2(0).z_o=4\times (2)^2+0^2-z_o^2$

$2_o^2=16^2$

$z_o=4$

From equation$(ii)$

$z=4+16(x-2)-8(y-0)$

$=4+16x-32-8y,$

$16x-8y-z-28=0$