(a) To determine Calculate the value of the Jacobian ∂ ( x , y ) ∂ ( u , v ) by solving the system in terms of u and v . Explanation Given information: The equation u is 3 x + 2 y . The equation v is x + 4 y . Calculation: Provide the Equation as follows: u = 3 x + 2 y (1) v = x + 4 y (2) Multiply both side by ( − 3 ) in Equation (2). − 3 v = − 3 x − 12 y (3) Add Equation (1) and (3). u − 3 v = − 10 y 10 y = 3 v − u y = 1 10 ( 3 v − u ) From Equation (1). u = 3 x + 2 y Substitute 1 10 ( 3 v − u ) for y . u = 3 x + 2 ( 1 10 ( 3 v − u ) ) = 3 x + 1 5 ( 3 v − u ) 3 x = u − 1 5 ( 3 v − u ) x = 1 5 ( 2 u − v ) Hence, the value of x and y in terms of u and v is 1 5 ( 2 u − v ) _ and 1 10 ( 3 v − u ) _ . Calculate the value of Jacobian as shown below: The Jacobian determinant or Jacobian of the coordinate transformation x = g ( u , v ) , y = h ( u , v ) . J ( u , v ) = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | (4) Find the partial derivatives as follows: Find ∂ x ∂ u : Differentiate x with respect to u . ∂ x ∂ u = ∂ ∂ u ( 1 5 ( 2 u − v ) ) = 2 5 Find ∂ x ∂ v : Differentiate x with respect to v (b) To determine Calculate the image under the transformation u and v of the triangular region in xy plane.

University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

4th Edition

ISBN: 9780135164860

Author: Joel R. Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.8, Problem 3E

(a)

To determine

Calculate the value of the Jacobian ∂(x,y)∂(u,v) by solving the system in terms of u and v.

(b)

To determine

Calculate the image under the transformation u and v of the triangular region in xy plane.

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Chapter 14.8, Problem 2E

Chapter 14.8, Problem 4E

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University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - Prob. 10E

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