# Instructor's Manual Applied Partial Differential Equations by Richard Haberman By Richard Haberman

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Additional resources for Instructor's Manual Applied Partial Differential Equations with Fourier Series and Boundary Value Problems Fourth Edition

Sample text

8. 12. 35 and note th a t x can be written i(x), where i is the identity linear transformation. 9. Define f : K2 —> R2 by f(x-, y) = (ex cosy, ex siily). a - Find f(0,ff/4)' ' b. Note th a t f(0,7r/4) = (y/2/2, \/2 /2 ); F in d ,f-1 '(v^ /2i>/ 2/2). c. 02) without computing f - 1 . Use Eq. 7) without the small r(h ) term . d. Apply f to your answer to P a rt (c) to see if it is reasonable. 10. Let z = f I ------- ), x ^ y. Show th at x —— h w— = 0. 6. 11. functions (u,v) — (e* + e2/,e:r — ey ) and (x,y) — ( in |-(« + t ’),ln £(u —w)) are inverses.

Show th at / has directional derivatives <9u/ ( x ) for all u at (0,0). Hint: Set u ■'■•i + sin#j). b. Show that. lim(Xi3/)_>(o,o) -f(x, y) does not exist, so / is discontinuous at (0,0). Hint: Compute / on the parabolas y = ax2. 4. Let f ( x , y ) — (x3 + J / ^ ) 3. Compute <)h/(0,0) from Eq. 23). 15. Why is this not a contradiction? 5. Let f (r,0) = (r c o s 0 ,r s in 0 ) . Compute . 6. -}. Find <9hf- Use SymPy. 7. Let g o f be the composition of f: ]Rm —> Rn followed by g : 1R” -> Mp. Suppose th at f is differentiable at; x and g at f(x).

If f(x) lias continuous partial derivatives in a open-set U C Rm. then f is continuously differentiable in U. 6, if f is continuously differentiable in U, then f is differentiable throughout IJ. (Confusing terminology, but it is standard. 8 '(Differential properties). Let a be a scalar and f and g be dif­ ferentiable vector valued functions defined on an open set U C I n . Then a. (af)x — af£. b- (f + g)x ~ C + gxThese follow from Theorem 3,2. 9. Let f : R m —> R ” be differentiable at x. Then, the matrix of the linear transformation f ' is rd fi dx\ dh d x\ dh -dxi d fi 1 dxm ' dh ■ ()j-m dh dx'2 Oh 8x2 O f 11 Ox 2 .