Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem
Multivariable Calculus Review
Multivariable Calculus Review
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Multivariable Calculus Review, Study Guides, Projects, Research of Calculus
Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norm. Multivariable Calculus Review.
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Multivariable Calculus Review
Multi-Variable Calculus
Point-Set Topology
Compactness
The Weierstrass Extreme Value Theorem
Operator and Matrix Norms
Mean Value Theorem
Multi-Variable Calculus
Norms: A function ν : Rn^ → R is a vector norm on Rn^ if I (^) ν(x) ≥ 0 ∀ x ∈ Rn^ with equality iff x = 0.
Multi-Variable Calculus
Norms: A function ν : Rn^ → R is a vector norm on Rn^ if I (^) ν(x) ≥ 0 ∀ x ∈ Rn^ with equality iff x = 0. I (^) ν(αx) = |α|ν(x) ∀ x ∈ Rn^ α ∈ R I (^) ν(x + y ) ≤ ν(x) + ν(y ) ∀ x, y ∈ Rn
Multi-Variable Calculus
Norms: A function ν : Rn^ → R is a vector norm on Rn^ if I (^) ν(x) ≥ 0 ∀ x ∈ Rn^ with equality iff x = 0. I (^) ν(αx) = |α|ν(x) ∀ x ∈ Rn^ α ∈ R I (^) ν(x + y ) ≤ ν(x) + ν(y ) ∀ x, y ∈ Rn
We usually denote ν(x) by ‖x‖. Norms are convex functions.
lp norms
‖x‖p := (
∑n i=1 |xi^ |
p (^) ) 1 p (^) , 1 ≤ p < ∞ ‖x‖∞ = maxi=1,...,n |xi |
Elementary Topological Properties of Sets
Let D ⊂ Rn.
Elementary Topological Properties of Sets
Let D ⊂ Rn. I (^) D is open if for every x ∈ D there exists > 0 such that x + B ⊂ D where x + B = {x + u : u ∈ B} and B is the unit ball of some given norm on Rn.
I (^) D is closed if every point x satisfying
(x + B) ∩ D 6 = ∅ for all > 0, must be a point in D.
Elementary Topological Properties of Sets
Let D ⊂ Rn. I (^) D is open if for every x ∈ D there exists > 0 such that x + B ⊂ D where x + B = {x + u : u ∈ B} and B is the unit ball of some given norm on Rn.
I (^) D is closed if every point x satisfying
(x + B) ∩ D 6 = ∅ for all > 0, must be a point in D.
I (^) D is bounded if there exists β > 0 such that
‖x‖ ≤ β for all x ∈ D.
Elementary Topological Properties of Sets: Compactness
Let D ⊂ Rn. I (^) x ∈ D is said to be a cluster point of D if
(x + B) ∩ D 6 = ∅
for every > 0.
Elementary Topological Properties of Sets: Compactness
Let D ⊂ Rn. I (^) x ∈ D is said to be a cluster point of D if
(x + B) ∩ D 6 = ∅
for every > 0.
I (^) D is compact if it is closed and bounded.
Continuity and The Weierstrass Extreme Value Theorem
The mapping F : Rn^ → Rm^ is continuous at the point x if
lim ‖x−x‖→ 0
‖F (x) − F (x)‖ = 0.
Continuity and The Weierstrass Extreme Value Theorem
The mapping F : Rn^ → Rm^ is continuous at the point x if
lim ‖x−x‖→ 0
‖F (x) − F (x)‖ = 0.
F is continuous on a set D ⊂ Rn^ if F is continuous at every point of D.
Operator Norms
A ∈ Rm×n^ ‖A‖(p,q) = max{‖Ax‖p : ‖x‖q ≤ 1 }
Operator Norms
A ∈ Rm×n^ ‖A‖(p,q) = max{‖Ax‖p : ‖x‖q ≤ 1 }
Examples: