# How do I apply Implicit function theorem correctly?

Reference:- An Elementary Course in Partial Differential Equations, T. Amaranth.

How do I apply implicit function theorem here?

My attempt:- I have gone through the generalized example of implicit function theorem from Principle of Mathematical Analysis by W. Rudin.

Here I understood that $$x_1=a,x_2=b, y_1=x,y_2=y,y_3=p,y_4=q.$$

My doubt:- I really unable to find $$f_1$$ and $$f_2$$ from the $$1.2.2,1.2.3$$ and $$1.2.4$$.

Suppose $$f_1(a,b,x,y,p,q)=p-F_x(x,y,a,b)$$ $$f_2(a,b,x,y,p,q)=q-F_y(x,y,a,b)$$

By Implicit function theorem, we have only $$2\times2$$ matrix $$\begin{pmatrix}p_{a}-F_{xa}&q_{a}-F_{ya}\\ p_{b}-F_{xb}&q_{b}-F_{yb}\end{pmatrix}.$$ I am not getting (1.2.5). I request you to find equations such that matrix in (1.2.5) has rank 2.

• Your $2\times 2$ matrix (once you fix typesetting typos) should have no $p,q$; remember that those are two of our variables. I think the book is wrong. They want to give you the option of using (1.2.2) instead of either (1.2.3) or (1.2.4). But we do not have a level set of $F$, and $z$ is not one of our variables. Commented Jul 23, 2022 at 18:59
• @TedShifrin may I know where did the author make mistake ? Commented Jul 24, 2022 at 1:21
• In the textbook, author illustrated two examples based on the matrix 1.2.5. Commented Jul 24, 2022 at 1:27
• The textbook is correct, but very telegraphic. I have written out a bit more of the details. They also have transposed the usual derivative matrix, which I find very annoying. Commented Jul 24, 2022 at 1:44

Let's try to rewrite this carefully. Define $$\Phi\colon\Bbb R^7\to\Bbb R^3$$ by $$\Phi(x,y,z,p,q,a,b) = \begin{bmatrix}z-F(x,y,a,b)\\p-F_x(x,y,a,b)\\q-F_y(x,y,a,b)\end{bmatrix}.$$ We want to consider the level set $$\Phi = 0$$. We want to consider any pair of these functions, not all three. The Implicit Function Theorem will guarantee that we can solve for $$a,b$$ as (smooth) functions of the remaining variables if the appropriate $$2\times 2$$ submatrix of the derivative matrix of $$\Phi$$ with respect to $$a,b$$ has rank $$2$$. Since $$\frac{\partial\Phi(x,y,z,p,q,a,b)}{\partial (a,b)} = -\begin{bmatrix} F_a & F_b \\ F_{xa} & F_{xb} \\ F_{ya} & F_{yb}\end{bmatrix},$$ their assertion is correct if this $$3\times 2$$ matrix indeed has rank $$2$$.