Question

Show that the notation {X_i} _i€I implicitly involves the notion of function.

Answer 1

If the equation F(x,y) =0 is written out in full terms of the coordinates, it will look like below

$F_1(x_1, ...,x_i, y_1, ...,y_i)=0$

$F_2(x_1, ...,x_i, y_1, ...,y_i)=0$

$.$

$.$

$.$

$F_i(x_1, ...,x_i, y_1, ...,y_i)=0.$

If $R^{i+1}\to R^i$ then d_xF is the differential of the function Rⁱ⁺¹ to Rⁱ obtained by fixing and

∂F/∂x is the corresponding Jacobi matrix , thus

$\frac{∂F}{∂ x} = \begin{bmatrix} \frac{∂F_1}{∂ x_1} & \frac{∂F_1}{∂ x_i} \\ \frac{∂Fi}{∂ x_1} & \frac{∂F_i}{∂ x_i} \end{bmatrix}$

Letting $R^{i+1}\to R^i$ be Cⁱ at a point (a,b) with F(a,b)=0. If d_xF(a,b) is invertible , there are positive numbers € and 𝛿

a) If $|x-a|< \delta$ then there is one and only one point $y= \varphi(x)$ satisfying $|x-b|< €$ and F(x,y)=0