Math 32AH Definitions

Cauchy Schwarz Inequality

(a dot b) <= |a||b|

Triangle Inequality

|a+b| <= |a| + |b|; |x-z| <= |x-y|+|y-z|; ||x|-|y|| <= |x-y|

Dot and Cross Product

a dot b = |a||b|costheta, a x b = |a||b| sintheta

Interior and Boundary Point of S

B(r,a) is in S → A is interior; B(r,a) meets both S and Sc —> A is boundary

Open and Closed Sets, S Closure

Open contain boundary points, closed contain none of the boundary points; S Closure is S U boundary of S

Epsilon-Delta Definition of a Limit

The limit at x = L, if for every epsilon > than 0, there exists delta s.t 0 < |x-a| < delta, and |f(x) - L| < epsilon.

Epsilon-Delta Definition of Continuity

If for every epsilon > 0, there exists delta s.t 0 < |x-a| < delta, and |f(x) - f(a)| < epsilon. Continuous at A if continuity exists at every point on A.

Basic Arithmetic Operations

Arithmetic operations of continuous functions are continuous except at dividing by 0.

Pre-Image Open Closedness

If the pre-image of an open set is open, or a closed set is closed, the function is continuous. PROVE THIS.

One Variable Differentiability & Mean-Value

f(a+h) = f(a) + mh + o(h); If f is differentiable and continuous on [a,b], there exists d in [a,b] s.t. f(a) - f(b) = f’(d) (a-b).

Gradient, Differentiability

lim h→0 (f(a+h) - f(a) - c dot h) / |h| = 0, Continuous partial derivatives imply differentiability

Tangent Plane

z= f(a,b) + ∂xf(a,b)(x−a) + ∂yf(a,b)(y−b), parameterizing (a,b,f(a,b))

Directional Derivative

lim t → 0 f(a + tu) - f(a) / t. If f is differentiable (all of it’s partials are continuous), the dertivative will also be equal to the gradient of f(a) dot u.

L is Linear if

L (sx + ty) = sL(x) + tL(y)

The Derivative Matrix

Df (a) = ∂1f1(a)··· ∂nf1(a)

∂1fm(a)··· ∂nfm(a)

equal to the gradient f(a)T for scalar valued a.

Chain Rule

D(f o g)(a) = Df(g(a)) Dg(a)

∂ℓHi(a) = sum from j = 1 to N of ∂jfi(g(a)) ∂ℓgj(a)

Homogenous Functions + Euler’s Identity

A function is positively homogenous of degree alpha if f(tx) = t ^ alpha f(x), for all t > 0 and x =/ to 0. Euler’s Identity states that x ·∇f(x) αf(x), if f is positively homogenous and differentiable.

Schwarz Theorem, The Hessian

Hf(a) = ∂i∂jf(a)n i,j=1

If the two mixed partials are continuous and all the partials exist, the two mixed partials can be taken in any order.

Multi-Index and Multinomial Formula

|α|= α1 +···+ αn, α! = α1!···αn!,

hα = h1^α1…hn^αn

(z1 + … zn)^r = sum of |α| = r r!/α! z^α

Convergence of a Sequence

A sequence Xk converges to a if for every epsilon > 0, there exists N s.t |Xk - a| < epsilon for all k >= N.

Accumulation Point

A point a is an accumulation point in S if every ball around a contains a point in S other than itself.

Completeness Axiom

Every nonempty subset of R that is bounded above has a supremum in R.

Compactness (Sequences Theorem) - PROVE

A subset of Rn is compact if it is closed and bounded. A set K ⊂ Rn is compact if and

only if every sequence in K has a convergent subsequence whose limit belongs to K.

Extreme Value Theorem

Let K ⊂Rn be nonempty and compact, and let f : K →R

be continuous. Then f attains its minimum and maximum: there are a,b ∈K such that

f(a) ≤f(x) ≤f(b) (x ∈K). 

Cauchy Sequences

A sequence is cauchy if for every epsilon > 0, there exists N s.t. |Xi - Xj| < epsilon for every i,j > N.

Bolzano Weierstrass

Every bounded sequence in Rn has a convergent subsequence.

Taylor’s Theorem

Lagrange’s Remainder

Quadratic Approximation (Taylor)

Taylor Polynomial