Problem #1: Let F: R² R², G: R2R2 be defined byr cos(0)r sin(0)F(r, 0)G(x, y) =2xyand let H = G. F. Using the chain rule, calculate the element of the Jacobian matrix DH(√2,27) correspondingto the first row and second column.

The differentiation of composite functionLet H=G∘F then H′=G′(F(r,θ))F′(r,θ)

Answered: = Let F: R2 R², G: R2-R2 be defined by…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.5: Solution Of Cubic And Quartic Equations By Formulas (optional)

Problem 29E

See similar textbooks

Similar questions

Explain what it means in terms of an inverse for a matrix to have a 0 determinant.
Consider f(t) = (cos t, sin t, t?) and g(x, y, z) = (x² + y² + z², 3x + 4y – 12z) and h = g(f(t)) = (h1, h2). The purpose of this problem is not to find the derivative of h . The purpose of this problem is not to find the first row of such derivative matrix either. The purpose of this problem is to find only the first entry of the first row of that derivative matrix. Which equation reflects that entry? dg de dg dy dy dt dg dz az dt dh where a = cos t, y = sint and z = t2 dt dx dt O dh2 dg dz dx dt where x = cost dt dh1 dg dæ where x = cost dt dx dt dg dæ dg dy dy dt dg dy dg dz az dt dh2 where x = cos t, y = sint and z = = t2 dt dx dt dh1 dg dæ dg dz dz dt where x = cos t, y = sint and z = = t2 dt dæ dt dy dt +
5) Find dx/dt of Matrix: X 5e-1 2e-¹ -7e-¹
Let f(u, v) = (u – v, 1,u + v) and g(x, y, 2) = xyz. Find the entry a12(i.e. the entry in the first row and second column) of the derivative matrices D/(u, v), Dg(x, y) and D(g • (0, 1). (a) a12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(g • )(0, 1) is (d) Select the correct answer about D(g • S(u, v) D(g • D(u, u) is a 2 x 3 matrix. D(g • (u, v) is a real-valued function of u and v. D(g • D(u, u) is a 1 x 2 matrix. D(g • D(u, u) is a 3 × 2 matrix. D(g • D(u, v) is a 2 x 2 matrix. (e) Select the correct answer about D(f • g)(x, y, z) D(S • g)(x, y, z) is not defined. D(S • g)(x, y, z) is a 3 x 2 matrix. D(S • g)(x, y, z) is a 2 x I matrix. DS • g)(x, y, z) is a 2 x 2 matrix. DS • g)(x, y, z) is a real-valued function of x and y. O O
7). Given functions F(x, y) = (x²y x³ – y + 4, ) and G(u, v) = (u² + v, v) compute the Jacobian derivative matrix of function G o F at the point (1,0).
Q6. Consider the function : R³ R² defined by f((x, y, z))= (-2x, y + 32). (a) Show directly that is linear. (b) Give the matrix associated to f. (c) Describe the set {ER | (7)-7).
Let f(u, v) = (u - v, 1,u + v) and g(x, y, z) = xyz. Find the entry a12(i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y) and D(g. f)(0, 1). (a) a12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(g)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) OD(g. f)(u, v) is a real-valued function of u and v. D(g. f)(u, v) is a 2 x 2 matrix. D(g. f)(u, v) is a 2 x 3 matrix. D(g f)(u, v) is a 3 x 2 matrix. 0 D(g f)(u, v) is a 1 x 2 matrix. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 2 x 1 matrix. D(f g)(x, y, z) is not defined. 0 D(f g)(x, y, z) is a 3 x 2 matrix. D(f g)(x, y, z) is a 2 x 2 matrix.
Evaluate |A|, where A is the 76 x 76 matrix: 1 2 3 4 76 ... 0 2 3 4 76 ... 0 0 3 4 76 ... A 0 0 0 4 76 ... 0 0 0 0 76 ... Note: The syntax for factorial is just x! ... ...
Are x, sin(x) and cox(x) linearly independent?
5. Given that G(x, y) = (x² + 1, y²) and F(u, v) = (u + v, v²), compute the Jacobian derivative matrix of F(G(x, y)) at the point (x, y) = (1, 1).
Let f(u, v) = (u - v, 1, u + v) and g(x, y, z) = xyz. Find the entry a12 (i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y, z) and D(gof)(0, 1). (a) a 12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(gof)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) D(gof)(u, v) is a 1 x 2 matrix. D(gof)(u, v) is a 2 x 2 matrix. D(gof)(u, v) is a 3 x 2 matrix. D(gof)(u, v) is a 2 x 3 matrix. D(gof)(u, v) is a real-valued function of u and v. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is not defined. D(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 3 x 2 matrix. D(fog)(x, y, z) is a 2 x 2 matrix. D(fog)(x, y, z) is a 2 x 1 matrix.
For geotrophic flows in the atmosphere of the Earth, the following first order system of ODE is formulated. X' = AX where 1 X2 and X= A = 1 X3 X4 The solution vector X relates to the velocities and accelerations of two perpendicular components of airflows tangential to the surface of the Earth. The symbols u and a represent the kinematic viscosity of air and the Coriolis constant; and both of them can be assumed as constants. (i) Find the eigenvalues of the matrix; (ii) Find the general solution of X. Useful formulas: X 3D АХ; det(А - 11) %3D 0; Х-с,х, +с, X, +......., X, and

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

SEE MORE TEXTBOOKS

= Let F: R2 R², G: R2-R2 be defined by r cos(0) r sin(8) F(r,0) = G(x, y) = 2xy and let H = G. F. Using the chain rule, calculate the element of the Jacobian matrix DH(√√2,27) corresponding to the first row and second column.