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October 10, 2007 MTHSC 974.001  Homework # 6 1. If we deﬁne f by f (x, y ) = (9−x2 −y 2 ) (3−x−y ) , 1 if x + y = 3 if x + y = 3 (6, 1) Is f continuous at (2, 1)? 2. If f : Df ⊆ Rm → Rn is continuous on Df and g : Dg ⊆ Rn → Rr is continuous on Dg Formulate and prove a theorem about when the composition f ◦ g is continuous. Be very clear about your statements! 3. Deﬁne f by f (x, y ) = Ax3 2 y 2 (x +y ) 0 where A is any nonzero constant. i.) Prove f is diﬀerentiable at all points (x, y ) ∈ R2 . ii.) Find
∂f ∂x if (x, y ) = (0, 0) if (x, y ) = (0, 0) and ∂f ∂y at all points (x, y ) ∈ R2 . iii.) Prove the four second order partials exist at all points (x, y ) ∈ R2 . iv.) Prove fxy (0, 0) = fyx (0, 0). 1 4. Let f be deﬁned on R2 by A(x2 + y 2 ) 0 where A is any nonzero constant. i.) Prove f is continuous only at (0, 0). ii.) Prove f is diﬀerentiable only at (0, 0). if (x, y ) ∈ QxQ else. f (x, y ) = 5. Let T : Rn → Rn be linear. Prove T is diﬀerentiable at all p ∈ Rn and T (p) = T for all p ∈ Rn . 6. Let T : Rn → Rn satisfy T (p) ≤ M · p Prove i.) T (0 ∈ Rn ) = 0 ∈ Rm . ii.) T is diﬀerentiable at 0 ∈ Rn . 2 for some positive M and all p ∈ Rn . iii.) T (0 ∈ Rn ) = the zero map from Rn → Rm . 7. Let T be the map from R3 → R3 deﬁned by T (r, θ, φ) = (x, y, z ) where T1 (r, θ, φ) = rsin(φ)cos(θ) T2 (r, θ, φ) = rsin(φ)sin(θ) T3 (r, θ, φ) = rcos(φ) (i.e.: the conversion from spherical coordinates in R3 to cartesian coordinates in R3 ). Let g (r, θ, φ) = r2 + 2θφ. Let F (r, θ, φ) = (gOT )(r, θ, φ) = x2 + 2yz = r2 sin2 (φ)cos2 (φ) + 2r2 sin(φ)cos(φ)sin(θ). Compute i.) ii.) iii.)
∂F ∂r ∂F ∂θ ∂F ∂φ both directly and by the chain rule. both directly and by the chain rule. both directly and by the chain rule. 2 ...
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This note was uploaded on 10/24/2009 for the course MTHSC 974 taught by Professor Peterson during the Fall '07 term at Clemson.
 Fall '07
 Peterson

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