Taylore Series of several variables Flashcards
Taylor series for a function of one variable
f(x) = f(a) + ∑(from k=1 to inf) f^(k)(a)/k! (x-a)^k
It can also be written as
f(x) = f(a) + ∑(from k=1 to inf) 1/k! (u ∂/∂x)^k f(a)
Taylor series for two variables
The taylor series for a two variable function f(x,y)=f(au, b+v), u=x-a, v=y-b is
f(a+u, b+v) = f(a,b) + ∑(from k=1, to inf) 1/k! (u ∂/∂x + v ∂/∂y) ^k f(a,b)
Taylor series for a three variables
The taylor series for a three variable function f(x,y,z) = f(a+u, b+v, c+w) = f(a,b,c) + ∑(k=1 to inf) 1/k! [ u ∂/∂x + v ∂/∂y + w ∂/∂z)^k f(a,b,c)
Tips for Taylor Series
Use known expansions where ever possible
Local maximum (of a one variable function)
Let f: D C |R^n -> |R. The point a e D is said to be a local maximum if f(x) =< f(c) for all x sufficiently close to a
Local minimum (of a one variable function)
Let f: D C |R^n -> |R. The point a e D is said to be a local minimum if f(x)>=f(c) for all x e D sufficiently close to a
Global maximum (of a one variable function)
Let f: D C |R^n -> |R. The point a e D is said to be a local maximum if f(x) =< f(c) for all x e D
Global minimum (of a one variable function
Let f: D C |R^n -> |R. The point a e D is said to be a local minimum if f(x)>=f(c) for all x e D
Local extremum
If it is a local minimum or maximum
Global extremum
If it a global minimum or maximum
Stationary or critical point
Let f: D C |R^n -> |R. The point a e D is a stationary or critical point if ▽f(a) =0
A singular point
Let f: D C |R^n -> |R. The point a e D is a singular point if ▽f does not exist at a.
Necessary conditions for extremum
Let D C |R^n -> |R. If f has a local extremum at the point a e D, then a must be either
(1) a stationary point of f or
(2) a singular point of f or
(3) a boundary point of D
Saddle point
A critical point a which is neither a local maximum nor a local minimum is called a saddle point
Hessian matrix (of 2 variables)
If f is a real function of two variables, and all the second order partial derivatives of f exist at the point (a,b) then the Hessian matrix of f at (a,b) is defined as
H= (fxx fxy)
(fyx fyy)
Hessian matrix (of 3 variables)
Hessian matrix of f at the point (a,b,c) is
H = (fxx fxy fxz)
(fyx fyy fyz)
(fzx fzy fzz)
The leading minor test (two variables)
Suppose f is a sufficiently smooth function of two variables with a critical point at (a,b) and H the Hessian matrix of f at (a,b) is denoted as follows
H=(fxx fxy) det(H) = fxx fyy - (fxy)^2
(fxy fyy)
where the partial derivatives in H are evaluated at (a,b)
The leading minor test (outcomes) (two variables)
If det(H) > 0 and fxx < 0, then (a,b) is a local maximum If det(H) > 0 and fxx > 0, then (a,b) is a local minimum If det(H) < 0, then (a,b) is a saddle point If det(H) =0, then we need to investigate further.
Leading minors of H
Let H be an nxn -matrix and for each 1≤ m≤n, let Hm be the mxm-matrix formed from the first m rows and m columns of H. The determinants det(Hm), 1≤m≤n are called the leading minors of H.
The leading minor test
If f: |R^n -> |R, then a stationary point x0 where det(H) ≠ 0 is
(1) a local maximum if (-1)^m det(Hm) >0 for all m, 1≤m≤n
(2) a local minimum if det(Hm) >0 for all m , 1≤m≤n
(3) a saddle point if none of the above.
If you cannot apply the test, you must classify a stationary point directly from the definition.
Boundedness Theorem
If f is a continuous function of n variables on a closed bounded domain of |R^n then f is bounded and attains its global maximum and minimum
Constrained extreme value problem
Maximise/minimise f(x,y)
Subject to the constraint g(x,y) = 0
Lagrangian function (Theory)
We can look for critical points of the Lagrangian function to find points on the curve g(x,y) =0 such that f(x,y) is maximum or minimum.
Lagrangian function
L(x, y, λ) = f(x,y) + λg(x,y)
Stationary equations of the Lagrangian function
At any critical point of L we must have
Lx = fx + λgx=0
Ly = fy + λgy=0
Lλ = g(x,y) = 0
This says that ▽f is parallel to ▽g.
