Explanation Given: The mapping G ( u , v ) = ( e u , e u + v ) Calculation: For one -one Suppose that G ( u 1 , v 1 ) = G ( u 2 , v 2 ) . We need to show that ( u 1 , v 1 ) = ( u 2 , v 2 ) ( e u 1 , e u 1 + v 1 ) = ( e u 2 , e u 2 + v 2 ) We get e u 1 = e u 2 which gives u 1 = u 2 and ( e u 1 + v 1 ) = ( e u 2 + v 2 ) gives u 1 + v 1 = u 2 + v 2 and hence v 1 = v 2 Thus G is one to one. The image of the exponent function is ( 0 , ∞ ) and x = e u , y = e u + v Here we can take any values for u and v We get x = e u > 0 , y = e u + v > 0 which means the image of G is first quadrant of the x y -plane and the axes are not included...

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.6, Problem 4E

To determine

To describe:

Let $G (u, v) = (e^{u}, e^{u + v})$

(a) Is $G$ one to one? what is the image of $G$ ?

(t) Describe the image of the vertical line $u = c$ and the horizontal line $v = c$ .

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Chapter 16.6, Problem 3E

Chapter 16.6, Problem 5E

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Chapter 16 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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Let G ( u , v ) = ( e u , e u + v ) (a) Is G one to one? what is the image of G ? (t) Describe the image of the vertical line u = c and the horizontal line v = c .

Solution Summary: The author explains that G is one to one. It is the first quadrant of the x y -plane with axes not included.

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To describe:

Let $G (u, v) = (e^{u}, e^{u + v})$

(a) Is $G$ one to one? what is the image of $G$ ?

(t) Describe the image of the vertical line $u = c$ and the horizontal line $v = c$ .

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Chapter 16 Solutions

Let G ( u , v ) = ( e u , e u + v ) (a) Is G one to one? what is the image of G ? (t) Describe the image of the vertical line u = c and the horizontal line v = c .

Solution Summary: The author explains that G is one to one. It is the first quadrant of the x y -plane with axes not included.

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To describe: Let G(u,v)=(eu,eu+v) (a) Is Gone to one? what is the image of G? (t) Describe the image of the vertical line u=cand the horizontal line v=c.

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Chapter 16 Solutions

To describe:

Let $G (u, v) = (e^{u}, e^{u + v})$

(a) Is $G$ one to one? what is the image of $G$ ?

(t) Describe the image of the vertical line $u = c$ and the horizontal line $v = c$ .