On a previous question, I got something related to linear algebra and linear algebra, but having no background in linear algebra and a little background in vector calculus(mainly from physics), I couldn't understand it. Can somebody explain this theorem in simpler terms, you need not explain all linear algebra:
Let $M$ be a diffrentiable manifold. Let $\bf G$ be a finite group of smooth maps from $M$ to $M$. Let P be a point in $M$ fixed by $\bf G$, and let $f$ be a diffrentiable function on $M$ invariant under $\bf G$. Assume that the action induced by $\bf G$ on the tangent space at $P$ is (nontrivial) and irreducible . Then $P$ is a critical point of $f$; and if this critical point is nondegenerate, it is a local maximum or minimum of $f$.
*I have marked words that I don't know with bold character.
What I could understand:
Maybe $M$ is some kind of vector with more than 3 components and it is diffrentiable(how can a vector be diffrentiable?).$\bf G$ is some kind of function from set $M$ to itself as if a function from $\mathbb R$ to itself. And $f$ is some function which is diffrentiable under $M$ and it remains unaffected by rotation of space, i.e. interchange of variables. As we have tangents on $2D$ graphs, we have some kind of tangent vectors onn multidimensional spaces. Maybe at $P$ the $f$ has a critical point(maybe where it's convexity changes, or a point of local maxima/minima).
Can someone guide me in right direction
This is not a linear algebra theorem. The study of differentiable manifolds is differential geometry, which relies on concepts from linear algebra.
From your questions, you appear to be a high-school student, probably studying calculus. Let me assure you that this theorem is quite advanced, and you're at least a few months of study away from understanding these terms.
I assume that you were looking for some theorem about maxima and minima of functions, as is standard in a calculus class approaching differentiation. What you have found is far, far more complicated than what you need. You should ask about what you're really trying to prove or learn.
Also, in your interpretation of the theorem, you appear to have more or less guessed at things. Why do you think that a finite group of smooth maps is the same as a function? I suggest not guessing at what advanced terms mean --- this may implant false knowledge in your mind, which you'll have to debug later on. Beyond that, you don't have to guess. There's a better way to interpret the theorem: look up the terms you don't understand. Great resources for this include: Wolfram Mathworld, Wikipedia, etc.