Let p = (5,2π, 1) ER³ and let ƒ : R³ →R be a C¹ function such thatafƒ(p) = 3, 3f (p) = 13,2x1af (p) = 7, af (p) = 7dx3əx2Calculate the trace of the Jacobian matrix tr(DF) evaluated at the point p for the function F : R³ → R³ defined byF(x) -=x² + x²-xx cos(9x₂)f(x)(Recall that the trace of a matrix is the sum of its diagonal entries.)

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Answered: Let p = (5, 2, 1) ER³ and let ƒ : R³ →…

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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7. Let T: R³ R³ be defined by (a) (b) (E) x1 + x2 + x3 = X1 X2 X3 [X1X2 X3] Find the matrix of linear transformation for T. T Find all vectors 7 € R³ such that T(x) = x.
Let A = M (f) be the matrix associated with the linear map ƒ((x,y,z)) = (x-2y,y+z,0) relative to the bases B={u= (1,0,3); v = (0,2,1), w = (2,1,0) } and B'= {u'=(0,0,2); v' = (1,0,3); w'=(1,3,1) } Then the first column of A is: O O 1/2 - 1/2 - 1/2 1/2 -1/2 1/2 - 1/2 1/2 -1/2
Define F(x, y, z)=(xyz, x² + yz,1+ 3x) for (x, y, z) in R³. Find the derivative matrix of the mapping F: R³ R³ at the points (1,2,3), (0,1,0), and (-1,4,0)
b 14. Compute the area of the portion of the saddle-like surface z = (2² - y²) that lies inside the cylinder x² + y² ≤ a². (Draw a sketch!) What is the leading-order term in this area as either a → 0 or b→ 0?
Let d : P5(R) → P5(R) denotes the first derivative. Show that d is a linear map. Choose a basis for P5(R) and find the matrix M(d). Use the matrix M(d) to find the derivative of f(x) = −9x^5 + 11x^4 −4x^2 +5x−1.
1. Let A=MB (f) be the matrix associated with the linear map f(x,y,z) = (3x-y + 4z,y+z-x,x+y+z) relative to the bases B={u=(4, 18,6), v= (17.1.0), w (17,0,-5) } and Then the first B'= {u'=(2,2,1); v' = (1,0,0); w'=(3,2,0) } column of A is: O A. 2 "B 1 3 B. - - 16 On () 13 1 C. D. E. 7 -5 3 4 17 17 3 -
Define T(f) = f′ from P4 to P4. (a) Prove T is linear. (b) Give the matrix for T. Is it invertible? Why or why not?
1. Let A = MB (f) be the matrix associated with the linear map f(x,y,z) = (2x − 2y+z, x − 4y+z, x-y-z) relative to the bases B={u=(5,16,1); v= (2, 1, –4); w=(7,17,2)} and B'={u'=(2,0,1); v' = (1,0,1); w'=(3,2,0)) Then the second column of A is: *) 0 O A. (0 O B. "B) °) SOC. O D. (6)
1. Let A = MB (f) be the matrix associated with the linear map f(x,y,z) = (3x−y+4z,y+z−x,x+y+z) relative to the bases B={u=(4,18,6), v= (17.1.0), w=(17,0,−5)} and B'={u'=(2,2,1); v'=(1,0,0); w'=(3,2,0) } Then the first column of A is: OA.-16 O B. 13 1 4 17 17 °C) O D. (2 213 O E. "G) ņ m

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