Assignment 1: Functions of several variables.

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1 Assignment 1: Functions of several variables. 1. State TRUE or FALSE giving proper justification for each of the following statements. (a) There eists a continuous function f : R R 2 such that f(cos n) = (n, 1 ) for all n N. n (b) There eists a non-constant continuous function f : R 2 R such that f(, y) = 5 for all (, y) R 2 with 2 + y 2 < 1. (c) There eists a one-one continuous function from (, y) R 2 : 2 + y 2 < 1} onto R 2. (d) There eists a continuous function from (, y) R 2 : 2 + y 2 1} onto R 2. (e) If f : R 2 R is continuous such that f (0, 0) eists, then f y (0, 0) must eist. (f) There eists a function f : R 2 R 2 which is differentiable only at (1, 0). (g) If f : R 2 R is continuous such that all the directional derivatives of f at (0, 0) eist, then f must be differentiable at (0, 0). [ ] 1 1 (h) If f : R 2 R 2 is differentiable with f(0, 0) = (1, 1) and [f (0, 0)] =, then there 1 1 cannot eist a differentiable function g : R 2 R 2 with g(1, 1) = (0, 0) and (f g)(, y) = (y, ) for all (, y) R 2. (i) A continuously differentiable function f : R 2 R 2 cannot be one-one and onto if det[f (, y)] = 0 for some (, y) R 2. (j) The equation sin(yz) = z defines implicitly as a differentiable function of y and z locally around the point (, y, z) = ( π, 1, 1). 2 (k) The equation sin(yz) = z defines z implicitly as a differentiable function of and y locally around the point (, y, z) = ( π, 1, 1) Let α (0, 1) and let n = (n 3 α n, 1 [nα]) for all n N. (For each R, [] denotes the n greatest integer not eceeding.) Eamine whether the sequence ( n ) converges in R 2. Also, find n if it eists. n 3. Eamine whether the following its eist and find their values if they eist. (a) (d) (,y) (0,0) 3 y 3 y 3 (b) (c) 4 + y 2 (,y) (0,0) 2 + y 2 (,y) (0,0) (,y) (0,0) y 2 e /y2 (e) 1 cos( 2 + y 2 ) (f) (,y) (0,0) ( 2 + y 2 ) 2 (,y) (0,0) 4. Eamine the continuity of f : R 2 R at (0, 0), where for all (, y) R 2, y if y 0, (a) f(, y) = y if y < 0. 1 if > 0 and 0 < y < 2, 0 otherwise. 3 (c) f(, y) = 2 +y 2 2 y 2 2 y 2 + ( 2 y 2 ) 2 2 y y 2 5. Determine all the points of R 2 where f : R 2 R is continuous, if for all (, y) R 2, y if y, (a) f(, y) = y 0 if = y. y if 2 y 2, 2 y 2 0 if 2 = y 2. y if y Q, (c) f(, y) = y if y R \ Q.

2 6. Let Ω be an open subset of R n and let f : Ω R m and g : Ω R m be continuous at 0 Ω. If for each ε > 0, there eist, y B ε ( 0 ) such that f() = g(y), then show that f( 0 ) = g( 0 ). 7. Let A( ) R n be such that every continuous function f : A R is bounded. Show that A is a closed and bounded subset of R n. 8. Let f : R 2 R be continuous at ( 0, y 0 ) R 2 and let f( 0, y 0 ) 0. Show that there eists δ > 0 such that f(, y) 0 for all (, y) R 2 satisfying ( 0 ) 2 + (y y 0 ) 2 < δ. 9. Let T : R n R n be linear and let f() = T () for all R n. Find f (), where R n. 10. Eamine the differentiability of f at 0, where (a) f : R n R satisfies f() 2 2 for all R n. (b) f : R 2 R is defined by f(, y) = y for all (, y) R 2. (c) f : R 2 R is defined by f(, y) = y y for all (, y) R 2. y (d) f : R 2 R is defined by f(, y) = y 2 + y 2 if y 0, (e) f : R 2 R is defined by f(, y) = (f) f : R 2 R is defined by f(, y) = (g) f : R 2 R 2 is defined by f(, y) = 0 if y = y 2 sin( 2 y 2 ) 2 +y 2 (sin sin 1, y2 ) if 0, (0, y 2 ) if = 0. (h) f : R n R is defined by f() = 2 for all R n. (i) f : R n R n is defined by f() = 2 for all R n. 11. Let f(, y) = ( 2 + y 3, 3 + y 2, 2 2 y 2 ) for all (, y) R 2. Eamine whether f : R 2 R 3 is differentiable at (1, 2) and find f (1, 2) if f is differentiable at (1, 2). 12. Determine all the points of R 2 where f : R 2 R is differentiable, if for all (, y) R 2, (a) f(, y) = 2 + y 2 if both, y Q, 0 otherwise. 4/3 sin( y ) if 0, 0 if = 0. ( 2 + y 2 1 ) sin( ) 13. Let f : R 2 R be defined by f(, y) = 2 +y2 Show that f is differentiable at (0, 0) although neither f nor f y is continuous at (0, 0). 14. Let f : R 2 R be defined by f(, y) = Eamine whether f y (0, 0) = f y (0, 0). 2 y( y) 2 +y 2

3 y( 2 y 2 ) 15. Let f : R 2 R be defined by f(, y) = 2 +y 2 Determine all the points of R 2 where f y and f y are continuous. 16. Let Ω be a nonempty open subset of R n. Let f : Ω R be differentiable at 0 Ω, let f( 0 ) = 0 and let g : Ω R be continuous at 0. Prove that fg : Ω R, defined by (fg)() = f()g() for all Ω, is differentiable at Let Ω be a nonempty open subset of R n and let g : Ω R n be continuous at 0 Ω. If f : Ω R is such that f() f( 0 ) = g() ( 0 ) for all Ω, then show that f is differentiable at The directional derivatives of a differentiable function f : R 2 R at (0, 0) in the directions of (1, 2) and (2, 1) are 1 and 2 respectively. Find f (0, 0) and f y (0, 0). 19. Find all v R 2 for which the directional derivative f v(0, 0) eists, where for all (, y) R 2, (a) f(, y) = 2 y 2. y 2 +y 2 if y 0, (c) f(, y) = y 0 if y = 0. (d) f(, y) = y y. 20. Let f(, y, z) = ( 3 y + y 2 z, yz) and g(, y) = ( 2 y, y, 2y, 2 + 3y) for all, y, z R. Use chain rule to find (g f) (a), where a = (1, 2, 3). 21. Let f : R 2 R be differentiable such that f(1, 1) = 1, f (1, 1) = 2 and f y (1, 1) = 5. If g() = f(, f(, )) for all R, determine g (1). 22. Let ϕ : R R and ψ : R R be differentiable. Show that f : R 2 R, defined by f(, y) = ϕ() + ψ(y) for all (, y) R 2, is differentiable. 23. Prove that a differentiable function f : R n \ 0} R m is homogeneous of degree α R (i.e. f(t) = t α f() for all R n \ 0} and for all t > 0) iff f ()() = αf() for all R n \ 0}. 24. Let f : R 2 R be continuously differentiable such that f (a, b) = f y (a, b) for all (a, b) R 2 and f(a, 0) > 0 for all a R. Show that f(a, b) > 0 for all (a, b) R Let Ω be an open subset of R n such that a, b Ω and S = (1 t)a + tb : t [0, 1]} Ω. If f : Ω R m is differentiable at each point of S, then show that there eists a linear map L : R n R m such that f(b) f(a) = L(b a). 26. Let f(, y) = (2ye 2, e y ) for all (, y) R 2. Show that there eist open sets U and V in R 2 containing (0, 1) and (2, 0) respectively such that f : U V is one-one and onto. 27. Determine all the points of R 2 where f : R 2 R 2 is locally invertible, if for all (, y) R 2, (a) f(, y) = ( 2 + y 2, y). ( 2 + y + y 2, y). (c) f(, y) = (cos + cos y, sin + sin y).

4 28. Determine all the points of R 3 where f : R 3 R 3 is locally invertible, if for all (, y, z) R 3, (a) f(, y, z) = ( + y, y + z, y + z). (b) f(, y, z) = ( y, y yz, yz). (c) f(, y, z) = ( 2 + y 2, y 2 + z 2, z ). 29. Let f(, y) = (3 y 2, 2 + y, y + y 3 ) and g(, y) = (2ye 2, e y ) for all (, y) R 2. Eamine whether (f g 1 ) (2, 0) eists (with a meaningful interpretation of g 1 ) and find (f g 1 ) (2, 0) if it eists. 30. For n 2, let B = R n : 2 < 1} and let f() = 2 2 for all B. Show that f : B B is differentiable and invertible but that f 1 : B B is not differentiable at Using implicit function theorem, show that the system of equations 3 (y 3 + z 3 ) = 0, ( y) 3 z 2 = 7, can be solved locally near the point (1, 1, 1) for y and z as a differentiable function of. 32. Show that the system of equations 2 y cos(uv) + z 2 = 0, 2 + y 2 sin(uv) + 2z 2 = 2, y sin u cos v + z = 0, implicitly defines (, y, z) as a differentiable function of (u, v) near = 1, y = 1, z = 0, u = π 2 and v = Using implicit function theorem, show that in a neighbourhood of any point ( 0, y 0, u 0, v 0 ) R 4 which satisfies the equations e u cos v = 0, v e y sin = 0, there eists a unique solution (u, v) = ϕ(, y) satisfying det[ϕ (, y)] = v/. 34. Show that in a neighbourhood of any point ( 0, y 0, z 0 ) R 3 which satisfies the equations 4 + ( + z)y 3 3 = 0, 4 + (2 + 3z)y 3 6 = 0, there is a unique continuous solution y = ϕ 1 (), z = ϕ 2 () of these equations. 35. Show that around the point (0, 1, 1), the equation y z log y + e z = 1 can be solved locally as y = f(, z) but cannot be solved locally as z = g(, y). 36. Show that the system of equations 3 + y z + u 2 = 0, y + 2z + u = 0, 2 + 2y 3z + 2u = 0,

5 can be solved locally for, y, u in terms of z; for, z, u in terms of y; for y, z, u in terms of, but not for, y, z in terms of u. 37. Show that the system of equations 2 + y 2 u 2 v = 0, 2 + 2y 2 + 3u 2 + 4v 2 = 1, defines (u, v) implicitly as a differentiable function of (, y) locally around the point (, y, u, v) = ( 1, 0, 1, 0) but does not define (, y) implicitly as a differentiable function of (u, v) 2 2 locally around the same point. 38. Show that there are points (, y, z, u, v, w) R 6 which satisfy the equations 2 + u + e v = 0, y 2 + v + e w = 0, z 2 + w + e u = 0. Prove that in a neighbourhood of such a point there eist unique differentiable solutions u = ϕ 1 (, y, z), v = ϕ 2 (, y, z), w = ϕ 3 (, y, z). If ϕ = (ϕ 1, ϕ 2, ϕ 3 ), find ϕ (, y, z). 39. Find the 3rd order Taylor polynomial of f(, y, z) = 2 y + z about the point (1, 2, 1). 40. Find the 4th order Taylor polynomial of g(, y) = e 2y /(1 + 2 y) about the point (0, 0). 41. Let f : R 2 R be continuously differentiable. Show that f is not one-one.