Let R^2 be equipped by the metric ds^2 = (4/(1 + x^2 + y^2 )^2) (dx^2 + dy^2 ), i.e. its first fundamental form is E = G = 4/(1+x^2+y^2)^2 , F = 0. Use the formula dω12 = −Kω1 ∧ ω2 to calculate its Gauss curvature.
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Let R^2 be equipped by the metric ds^2 = (4/(1 + x^2 + y^2 )^2) (dx^2 + dy^2 ), i.e. its first fu...
2. Let (X, dx), (Y, dy) be two metric spaces, and f:X + Y a map....
2. Let (X, dx), (Y, dy) be two metric spaces, and f:X + Y a map. (a) Define what it means for the map f to be continuous at a point x E X. (b) Suppose W X is compact. Prove that then f(W) CY is compact.
2. Let (X, dx), (Y, dy), (2, dz) be metric spaces, and f : XY,g:Y +...
2. Let (X, dx), (Y, dy), (2, dz) be metric spaces, and f : XY,g:Y + Z continu- ous maps. (a) Prove that the composition go f is continuous. (b) Prove that if W X is connected, then f(W) CY is connected.
Consider the set of integers Z with the metric da.y)-2supm e Nu (o): 2" divides (r-y)...
Consider the set of integers Z with the metric da.y)-2supm e Nu (o): 2" divides (r-y) (a) Describe the open balls of radius 1 around the centres 0 and 1 (b) Let f : Z -R be defined as f()0 if is even and (x)1 f r is odd. Is f a continuous function from (Z, d) to R equipped with the standard metric? Himt: Use the criterion of continuity in terms of open sets
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 ...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y....
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y. Does lim, fn(x) exist for all E R? If it exists, is the convergence uniform. Justify 2. Consider fn(x) = x2m, x E [0, 1]. Is it true that lim (lim fn(= lim( lim fn(x)) noo x-1 Justify.
2. Let f(x,y) = e In(y) and let R be the region in the first quadrant...
2. Let f(x,y) = e In(y) and let R be the region in the first quadrant of the plane that lies above r = = In(y) from y=1 to y = 2. (a) Sketch the region R in the plane. (b) Evaluate SSR f(x,y) dA.
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
(1) Evaluate the integral (r-1) min(a, y) dy dr, Jo Jo where min(x, y) is the minimum value of r ...
q2 please
(1) Evaluate the integral (r-1) min(a, y) dy dr, Jo Jo where min(x, y) is the minimum value of r and y. (2) Let f,g : R → R be functions of one variable such that f" and g" are continuous. Show that (f"(x)-g"(y)) dydx = f(0) + g(0)-f(2)-9(2) + 2f'(2) + 2g'(0). o Jo (3) Let a > 0. In spherical coordinates, a surface is defined by r = 2acos φ for 0 φ 1. Find the...
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be...
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be a limit point of X, and let yo e Y be a limit point of Y. Let f : X+Y be a function such that f(xo) = yo, and such that f is differentiable at Xo. Suppose that g:Y + R is a function which is differentiable at yo. Then the function gof:X + R is differentiable at xo, and .. (gºf)'(xo) = g'(yo)...
2. Let (X, dx), (Y, dy) be two metric spaces, and f:X + Y a map. (a) Define what it means for the map f to be continuous at a point x E X. (b) Suppose W X is compact. Prove that then f(W) CY is compact.
2. Let (X, dx), (Y, dy), (2, dz) be metric spaces, and f : XY,g:Y + Z continu- ous maps. (a) Prove that the composition go f is continuous. (b) Prove that if W X is connected, then f(W) CY is connected.
Consider the set of integers Z with the metric da.y)-2supm e Nu (o): 2" divides (r-y) (a) Describe the open balls of radius 1 around the centres 0 and 1 (b) Let f : Z -R be defined as f()0 if is even and (x)1 f r is odd. Is f a continuous function from (Z, d) to R equipped with the standard metric? Himt: Use the criterion of continuity in terms of open sets
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y. Does lim, fn(x) exist for all E R? If it exists, is the convergence uniform. Justify 2. Consider fn(x) = x2m, x E [0, 1]. Is it true that lim (lim fn(= lim( lim fn(x)) noo x-1 Justify.
2. Let f(x,y) = e In(y) and let R be the region in the first quadrant of the plane that lies above r = = In(y) from y=1 to y = 2. (a) Sketch the region R in the plane. (b) Evaluate SSR f(x,y) dA.
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
q2 please
(1) Evaluate the integral (r-1) min(a, y) dy dr, Jo Jo where min(x, y) is the minimum value of r and y. (2) Let f,g : R → R be functions of one variable such that f" and g" are continuous. Show that (f"(x)-g"(y)) dydx = f(0) + g(0)-f(2)-9(2) + 2f'(2) + 2g'(0). o Jo (3) Let a > 0. In spherical coordinates, a surface is defined by r = 2acos φ for 0 φ 1. Find the...
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be a limit point of X, and let yo e Y be a limit point of Y. Let f : X+Y be a function such that f(xo) = yo, and such that f is differentiable at Xo. Suppose that g:Y + R is a function which is differentiable at yo. Then the function gof:X + R is differentiable at xo, and .. (gºf)'(xo) = g'(yo)...
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