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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. 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. T h e A d v e n t u r e s o f T o m S a w y e r p S t r o n g D e s i r e — W i s e A c t i o n S ATURDAY MORNING CAME A LL THE SUMMER WORLD WAS BRIGHT and fresh and full of life Tom appeared in front of the house with paint and a big brush He looked at the fence, and all joy left him A deep sadness settled upon his heart.

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