Please give me an intuitive explanation of ‘implicit function theorem’. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why they use Jacobian matrix to illustrate this theorem.
Let’s use a simple example with only two variables. Assume there is some relation f(x,y)=0 between these variables (which is a general curve in 2D). An example would be f(x,y)=x2+y2−1 which is the unit circle in R2. Now you are interested to figure out the slope of the tangent to this curve at some point x0,y0 on the curve [with f(x0,y0)=0].
What you can do is to change x a little bit x=x0+Δx. You are interested then how y changes (y=y0+Δy); remember that we are interested in points on the curve with f(x,y)=0. Using Taylor expansion on f(x,y)=0 yields (up to lowest order in Δx and Δy)
The slope is thereby given by ΔyΔx=−∂xf(x0,y0)∂yf(x0,y0). As Δx→0 higher order terms in the Taylor expansion (which we neglected) vanish and ΔyΔx becomes the slope of the curve implicitly defined via f(x,y)=0 at (x0,y0).
More variables and higher dimensional spaces can be treated similarly (using Taylor series in several variables). But the example above should provide you with enough intuition and insight to understand the ‘implicit function theorem’.