Week 16 Flashcards
(16 cards)
How to describe e.g. f(x,y)
or f(x1,x2,x3)
f(x,y) = 2d hypersurface in R^3
f(x1,x2,x3)= 3d hypersurface in R^4
Tangent hyperplane eq?
Same as tangent plane but swap the LHS terms
Vector paramteric eq for tangent hyperplane?
same as tangent plane with partial derivatives and 1 0 vectors
Derivative of …
f(t) = F(x(t)y(t))
f(x)=F(x y(x))
f’(t) = Fx((x(t)y(t)))x’(t) + Fy((x(t)y(t)))y’(t)
f’(x) = Fx((x y(x)) + Fy((x y(x)))y’(x)
Derivative of
f(x,y(x,z),z)
so fx(x,y(x,z),z)
fz(x,y(x,z),z)
= Fx(x,y(x,z),z) + Fx(x,y(x,z),z) y x(x,z) <last term is p.d(x)
=Fz(x,y(x,z),z) + Fz(x,y(x,z),z) y z(x,z) <last term is p.d(z)
How to find stationary points of f: R^n -> R?
All partial derivatives = 0
f’‘(a) matrix?
fx1x1(a) fx1x2(a) fx1x3(a) …
fx2x1(a)…….
fx3x1(a) …… (symetrical)
P2(a) (taylor polynomial matrix?
P2(a) = f(a) + f’(a)(x-a) + 1/2(x-a)^T f’‘(a)(x-a)
Eigenvalues test for SP?
strict local min if f’‘(a) is positive definite (all eigenvalues are >0)
strict local max if f’‘(a) is negative definite (all eigenvalues are<0)
saddle point if indefinite e.g. one positive and one negative eigenvalue
Prinicipal minors what are they?
det of f’‘(a) top left then increasing boxes e.g. 1x1 then 2x2 then 3x3
Odd principle minors?
det of 1x1 3x3 e.g.
Even principle minors?
det of 2x2 4x4 e.g.
Principle minors test for SP?
local min (positive def) if all minors > 0
local max (negative def) if all even minors >0 and odd < 0
indefinite and saddle point if neither and det f’‘(a) ≠ 0
How to tell if function is convex or concave?
If ALL eigenvalues are positive semi definite ≥ 0 (then any local min is a global min) CONVEX
If ALL eigenvalues are negative semi definite ≥ 0 (then any local min is a global max)
(CONCAVE)
Ad hoc method for finding if a point is global extrema?
find the value using sub of s.p and just get an eq that has less than or more than
and if no soln’s then global extrema
Plane tangent to surface?
other words for tangent line eq but for 3d
so you do the horizontal thing where you make the LHS=0
