# Multivariable calculus - Implicit function theorem

we are given the function $F: \mathbb R^3 \to \mathbb R^2$, $F(x,y,z)=\begin{pmatrix} x+yz-z^3-1 \\ x^3-xz+y^3\end{pmatrix}$

Show that around $(1,-1,0)$ we can represent $x$ and $y$ as functions of $z$, and find $\frac{dx}{dz}$ and $\frac{dy}{dz}$

What I did:

The differential of $F$ is:

$\begin{pmatrix} 1 & z &y-3z^2\\3x^2-z & 3y^2 &-x\end{pmatrix}$, insert $x=1,y=-1,z=0$ to get:

$\begin{pmatrix} 1 & 0 &-1 \\3&3&-1\end{pmatrix}$

The matrix of the partial derivatives with respect to x and y is the first 2 columns, and it is invertible, and so the requirements of the implicit function theorem are met.

How do i find the differential of $x$ and $y$ with respect to $z$ tho?

One would expect that $\frac{dx}{dz} = -\frac{dF}{dz}(\frac{dF}{dx})^{-1}$ but...those are vectors. what is the inverse of a vector? how do you multiply vectors? there's a size mismatch.

• The formula $\frac{dx}{dz} = -\frac{dF}{dz}(\frac{dF}{dx})^{-1}$I'm guessing you took it from a theorem that was presented to you or something. If that's the case, can you state the theorem? – Git Gud Apr 5 '14 at 17:26
• I'm not sure, I think it's the chain rule, but in any way, it's the exact same thing you did here math.stackexchange.com/questions/731725/… I thought it might work in other examples as well. I have a big problem with this subject, I also have another open question where my problem is finding the derivative – Oria Gruber Apr 5 '14 at 18:43

The implicit function theorem: Let $$m,n$$ be natural numbers, $$\Omega$$ an open subset of $$\mathbb R^{n+m}$$, $$F\colon \Omega\to \mathbb R^m$$ a class $$C^1$$ function and $$(a_1, \ldots ,a_n, b_1, \ldots ,b_m)\in \Omega$$ such that $$F(a_1, \ldots ,a_n, b_1, \ldots ,b_m)=0_{\mathbb R^{\large m}}.$$ Writing $$F=(f_1, \ldots, f_m)$$ where for each $$k\in \{1, \ldots m\}$$, $$f_k\colon \mathbb R^{n+m}\to \mathbb R$$ is a class $$C^1$$ function, assume that the following $$m\times m$$ matrix is invertible: $$\begin{pmatrix} \dfrac{\partial f_1}{\partial y_1} & \cdots & \dfrac{\partial f_1}{\partial y_m}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial y_1} & \cdots& \dfrac{\partial f_m}{\partial y_m}\end{pmatrix}(a_1, \ldots ,a_n, b_1, \ldots ,b_m).$$

In these conditions there exists a neighborhood $$V$$ of $$(a_1, \ldots ,a_n)$$, a neighborhood $$W$$ of $$(b_1, \ldots ,b_m)$$ and a class $$C^1$$ function $$G\colon V\to W$$ such that:

• $$G(a_1, \ldots ,a_n)=(b_1, \ldots ,b_m)$$ and
• $$\forall (x_1, \ldots ,x_n)\in V\left(F(x_1, \ldots , x_n, g_1(x_1, \ldots , x_n), \ldots ,g_m(x_1, \ldots , x_n))=0_{\mathbb R^{\large m}}\right)$$, where for each $$l\in \{1, \ldots , m\}$$, $$g_l\colon \mathbb R^n \to \mathbb R$$ is a class $$C^1$$ function and $$G=(g_1, \ldots ,g_m)$$.

Furthermore, $$J_G=-\left(J_2\right)^{-1}J_1$$ where $$J_G \text{ is } \begin{pmatrix}\dfrac {\partial g_1}{\partial x_1} & \cdots & \dfrac {\partial g_1}{\partial x_n}\\ \vdots &\ddots &\vdots\\ \dfrac {\partial g_m}{\partial x_1} & \cdots & \dfrac {\partial g_m}{\partial x_n} \end{pmatrix}_{m\times n}\\ \text{ evaluated at }(x_1, \ldots ,x_n), \\~\\ J_2\text{ is }\begin{pmatrix} \dfrac{\partial f_1}{\partial y_1} & \cdots & \dfrac{\partial f_1}{\partial y_m}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial y_1} & \cdots& \dfrac{\partial f_m}{\partial y_m}\end{pmatrix}_{m\times m}\\ \text{ evaluated at }(x_1, \ldots , x_n, g_1(x_1, \ldots , x_n), \ldots ,g_m(x_1, \ldots , x_n)),$$ and $$J_1\text{ is }\begin{pmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial x_1} & \cdots& \dfrac{\partial f_m}{\partial x_n}\end{pmatrix}_{m\times n}\\ \text{ evaluated at }(x_1, \ldots , x_n, g_1(x_1, \ldots , x_n), \ldots ,g_m(x_1, \ldots , x_n)).$$

In this problem we can't apply the IFT as it is, because to use this version of the IFT one writes the last $$m$$ variables as functions of the first $$n$$ ones, but looking at the proof one notices that we can just consider permutations of this and this is what happens here.

In the notation above one has $$n=1, m=2, \Omega =\mathbb R^{n+m}, F\colon \mathbb R^{n+m}\to \mathbb R^m$$ given by $$F(x,y,z)=(f_1(x,y,z), f_2(x,y,z))$$, where $$f_1(x,y,z)=x+yz-z^3$$ and $$f_2(x,y,z)=x^3-xz+y^3$$.

For all $$(x,y,z)\in \mathbb R^3$$ it holds that:

• $$\dfrac {\partial f_1}{\partial x}(x,y,z)=1,$$
• $$\dfrac {\partial f_1}{\partial y}(x,y,z)=z,$$
• $$\dfrac {\partial f_2}{\partial x}(x,y,z)=3x^2-z,$$ and
• $$\dfrac {\partial f_2}{\partial y}(x,y,z)=3y^2$$.

Therefore $$\begin{pmatrix} \dfrac {\partial f_1}{\partial x}(1,-1, 0) & \dfrac {\partial f_1}{\partial y}(1, -1, 0)\\ \dfrac {\partial f_2}{\partial x}(1, -1, 0) & \dfrac {\partial f_2}{\partial y}(1, -1, 0)\end{pmatrix}=\begin{pmatrix} 1 & 0\\ 3 & 3\end{pmatrix}$$ and the matrix $$\begin{pmatrix} 1 & 0\\ 3 & 3\end{pmatrix}$$ is invertible.

So, by the IFT, there exists an interval $$V$$ around $$z=0$$ and a neighborhood $$W$$ around $$(x,y)=(1,-1)$$ and a class $$C^1(V)$$ function $$G\colon V\to W$$ such that $$G(0)=(1-1)$$ and $$\forall z\in V\left(F(g_1(z), g_2(z), z)=0\right)$$, where $$g_1(z), g_2(z)$$ denote the first and second entries, respectively, of $$G(z)$$, for all $$z\in V$$. (In analyst terms, $$g_1(z)=x(z)$$ and $$g_2(z)=y(z)$$).

One also finds

$$\begin{pmatrix} \dfrac {\partial g_1}{\partial z}(z)\\ \dfrac {\partial g_2}{\partial z}(z) \end{pmatrix}=\\ -\begin{pmatrix} \dfrac{\partial f_1}{\partial x}(g_1(z), g_2(z), z) & \dfrac{\partial f_1}{\partial y}(g_1(z), g_2(z), z)\\ \dfrac{\partial f_2}{\partial x}(g_1(z), g_2(z), z) & \dfrac{\partial f_2}{\partial y}(g_1(z), g_2(z), z) \end{pmatrix}^{-1} \begin{pmatrix} \dfrac{\partial f_1}{\partial z}(g_1(z), g_2(z), z)\\ \dfrac{\partial f_2}{\partial z}(g_1(z), g_2(z), z) \end{pmatrix}_.$$

Now you can happily evaluate the RHS at $$z=0$$.

If the variables are $(u,v)$ for $\mathbb{R}^2$ then the map is $$u=x+yz-z^2+1,\\ v=x^3-xz+y^3.\tag{1}$$ Note at the point $(x,y,z)=(1,-1,0)$ these coordinates are $(u,v)=(0,0).$ If it is possible to get $x,y$ implicitly in terms of $z$, then it must be that the variables in $(1)$ are both put to the constant $0$, and then each equation brings the dimension down by one, so the result is a one dimensional part of 3-space. This means one can take both derivatives w.r.t. $z$ and set them to $0$, and get a simultaneous system involving $dx/dz$ and $dy/dz$.

$$(dx/dz)+y+(dy/dz)z-2z=0,\\ 3x^2(dx/dz)-(dx/dz)z-x+3y^2(dy/dz)=0.$$

After moving terms to the right side not involving $dx/dz,\ dy/dz,$ the linear system in terms of $A=dx/dz,\ B=dy/dz$ is $$1 A +z B=-y+2z,\\ (3x^2-z)A+(3y^2)B=x.$$ This system can now be solved for $A,B$ using Cramer's rule.