3. Let f(x)=x² + 3 and g(x)=x³-1. Use the standard inner product on C[0, 1] to compute thefollowing:(a) d(f, g)(b) ||g||(c) the projection of fonto g

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Answered: 3. Let f(x)=x² + 3 and g(x)=x³-1. Use…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Letf : R² → R be defined by ƒ((x, y)) = −3x – 7y. Is ƒ a linear transformation? a. f((x₁, y₁) + (x₂, y2)) = f((x₁, y₁)) + f((x₂, y2)) = Does f((x₁, y₁) + (x₂, y₂)) = f((x₁, y₁)) + f((x₂, y₂)) for all (x₁, y₁), (x₂, y₂) E R²? choose . (Enter x₁ as x1, etc.) b. f(c(x, y)) = c(f((x, y))) = = Does f(c(x, y)) = c(f((x, y))) for all c E R and all (x, y) = R²? choose c. Isf a linear transformation? choose
(4) Let f₂ = 0ƒ/əz and ƒz = 0ƒ/az. Recall f = u + iv, the notation f is used for the complex conjugate function u – iv. (e) Show that if ƒ = O(U), then A|f² = 4|f¹|².
Letf: R2. →R be defined by f((x, y)) = 4y - 3x. Is f a linear transformation? a. f((x₁, y₁) + (x2, y2)) = b. f(c(x, y)) = f((x₁, y₁)) + f((x2, y2)) = + Does f((x1, y1) + (x₂, y₂)) = f((x₁, y₁)) + f((x₂, y₂)) for all (x₁, y₁), (x2, y₂) E R²? choose (Enter x₁ as x1, etc.) c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c ER and all (x, y) E R²? choose c. Is f a linear transformation? choose ⇒
part D
(d) Demonstrate mathematical reasoning to prove that that the real-valued function x2 – y2 u(x, y) = x3 + 2x² – 3xy2 – 2y² + (x2 + y2)2 is harmonic. Construct a harmonic conjugate of u(x, y) and write down an entire function f(z) such that Re(f(z)) = u(x, y), where z = x + yi. In your working, you may use the integration formula a? – 3y? y (a2 + y?)3 dy (a? + y2)² + C for any real constant a.
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
Let f and g be analytic functions in a domain D. Iff '(z) =g'(z)for all z in D,then show thatf (z) = g(z) + c, where c is a complex constant.
(b) Given Â = (x²y)î + (xz – z²)f – (xyz)k, determine the following at point (1, 1, 2): (1) V - Â. (ii) grad(div Â). (iii) curl(curl Â).
Use the inner product
Let 3. f(2) = z² – 2iz be a complex valued function defined on a complex plane. Writing z = x + iy, 6. find real valued functions u(x, y) and v(x, y) such that f(2) = u(x, y) + iv(x, y). u(x, y) = 2x² + 2y? + x and v(x, y) = 4xy – Y u(x, y) = x² – y² + 2y and v(x, y) = 2xy – 2x u(x, y) = 2x ² – 2y + y and v(x, y) = 4xy- x u(x, y) = x2 +y2 +x and v(x, y) = 2xy- y
Consider the utility function u'(x1, X2, X3) = x²1 x³2 X3 . Find a monotone transformation of u'(x1, X2, X3) such that the new utility function is of the form u2(x1, X2, X3) = (X1º, X2º, x3º) Where a + b + c= 1.
3. Let f: AB and g: BC be invertible functions. Prove that (gof): A+C is invertible and (gof)-¹=f-¹ og ¹.

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3. Let f(x)=x² + 3 and g(x)=x³-1. Use the standard inner product on C[0, 1] to compute the following: (a) d(f, g) (b) || g || (c) the projection of fonto g