2. Prove or disprove these two statements: (i) if ƒ is Riemann integrable on [a, b] then so is |ƒ|. (ii) if f is Riemann integrable on [a, b] then so is f. (Hint: Try a slight modification of the Dirichlet function as a potential counterexample).

Q: 2) Let F= 2 xyi + (x² + 2) j. find a potential function, f, for it. Show that Fis conservative and

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Q: 2. Consider the floor function [.] whose output is the largest integer less than or equal to the…

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Q: Show that the operator U is positive U: 1² ([0,1]) → L²([0,1]), Uƒ (x) = X[{0,1] [~*\f(t)\dt

A: To show that the operator U is positive, need to demonstrate that for any f∈L2([0,1]), the inner…

Q: analytic function a domain D. If a zeck. y) + b v(x, y) = C in D are real Constants not all o. #2.…

A: If f(z)=u+iv is analytic in a domain D , then f satiesfies the Cauchy Riemann (C-R) equations that…

Q: 1. Consider the vector-valued function F(t) = sin(2t) t (2, 1, -2), a) Find the domain of F. b) Show…

A: Disclaimer: There are two questions here and as per Bartleby policy, I am solving only the first…

Q: A vector field F = x(x² – y²) + ŷ(2xy + y?) a) f d.č =? c on the closed curve in the figure b)…

A: a) Consider the figure:

Q: 2. Consider the vector-valued function R(t) = { Show that is continuous at t = 1. (el-¹, 1-², In (2-…

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Q: State what it means for a scalar function y = f (x) to beintegrable on an interval [a, b].

A: Scalar integrable  function y = f (x)

Q: 3. Let f(z) be an analytic function in D = B₁(0) = {x² + y² < 1} and f(z)| = 1. Show that f(2)= C.…

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Q: The nabla formulas corresponding to the derivative rule of income are gVf+fVg, (Vf). F + f(V.F),…

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Q: Find the analytic function f (z)u+iv if u = a (1+cose).

A: We have to find the analytic function

Q: [#1] Exercise 7.46. Prove that if y = y' then y = ky' for some scalar k.

A: Given that γ≡γ' The y can be written as,  γ≡r mod n The y' can be written as, γ'≡ r mod n

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: Show that Vx (V.f) = 0 for any vector field f.

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Q: 2.) Are the following functions vector-valued or scalar-valued? a) f(u,v, w) = Cu²v, we, sv) b.j…

A: Answer given below.

Q: Use the Cauchy-Riemann equations to show that the function f(z) - e² is not analytic anywhere.

A: The solution is given as

Q: Let z = x+iy and f(2)=√√|xy|. Show that ƒ(z) satisfies the Cauchy-Riemann equations at the origin,…

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Q: Let f : R" → R be a continuously differentiable function, while æ € R" is its non-stationary point…

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Q: 1 Justify the Cauchy-Riemann equations for f(2) = e² where f(z) is holomorphic.

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Q: If r and u are differentiable vector-valued functions of t, then r(t) - u(t)) = r'(t) - u'(t). O…

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Q: Show that V x VG = 0 for all scalar field G(x,y,z). %3D

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Q: #2. Let flz) = (x, y) +iv(x,y) be an analytic function in a domain D. If a zeck. y) + b v(x, y) = C…

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Q: 5. Find all values of t for which the vector-valued function is continuous: r(t) = Inti + ₁ + √t-2k…

A: Use the given information.

Q: Q.3) Graph the vector field F(x, y) = –yi. Find V.(F × G) where F(x, y, z) = xi – 3j+ zk and G(r, y,…

A: given vector field Fx,y=-yi^  graph of the vector field

Q: Prove or disprove: f is invertible. 3. (a) f: R→ R, f(x) = 3x.

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Q: a) Verify the Cauchy-Riemann equations and find f'(z) for f(z) = z2 , f(z) =| z |2 and f(z) = 1

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Q: 4. Suppose that f and g are Riemann integrable functions on [a, b]. (a) Show that 1/2 1/2 ||…

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