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2 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all x. Dean te dea ns g a t e c ross street r o s s s t r e e t ch ur c h st c h u r h e s t dale streetd a l e grosvenor st s t r e e t dale streetd a l e s t r ee t oldham. ̂ ߁A R s A j V Ŏg p 邩 L N ^ A j A V ɃX N Z o Ƃ Ď ^ ׂĂ̊ ̕ ɁA y ł 悤 Ɏd l ύX ܂ B.
Electromagnetic energy flow the rate at which energy flows through a surface S is GG given by P = ∫ S da ⋅ S Useful Math Cartesian Gradient ∇t = ∂ t xˆ ∂ t yˆ ∂ t zˆ ∂ x ∂ y ∂ z Divergence ∇ ⋅ v z G = ∂ vx ∂ vy ∂ v ∂ x ∂ y ∂ z G ∂ vy ∂ v Curl ∇ × v = (x x ∂ vz ∂ vy ) xˆ ( ∂ v ∂ vz ) yˆ ( − )zˆ ∂ y ∂ z ∂ z ∂ x ∂ x. T E s y E s ̖ C X g E J b g f. C X t H X g C A L b g ň l łf d s I S z ԋ ۏ t Ă I I ɔ Ă 郉 C X t H X g C A L b g ͂ B.
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Eƒ Ae A A ˆa E Sa A A Ae A C Aœ E œas Ae E Az A A A Ae C
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Eƒ Ae A A ˆa E Sa A Zae A A A Ae Aˆ Ae E C Ae E Az A A A Ae C
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