bmhicks_homework_6

# bmhicks_homework_6 - Brandy Hicks October 10, 2007 MTHSC...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Brandy Hicks 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 ...
View Full Document

## This note was uploaded on 10/24/2009 for the course MTHSC 974 taught by Professor Peterson during the Fall '07 term at Clemson.

### Page1 / 2

bmhicks_homework_6 - Brandy Hicks October 10, 2007 MTHSC...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online