Let h be the vector field h(x, y, z) = (2xy + ²)i + (x² − 2yz)j + (2x= − y²) k. (a) Show that h is a gradient field. (b) What is the value of [((2xy + 2²) dx + (x² − 2yz) dy + (2x= − y²) dz for every piecewise-smooth curve C (i) from (1, 0, 1) to (3, 2, -1)? (ii) from (3, 2, -1) to (1, 0, 1)?
Let h be the vector field h(x, y, z) = (2xy + ²)i + (x² − 2yz)j + (2x= − y²) k. (a) Show that h is a gradient field. (b) What is the value of [((2xy + 2²) dx + (x² − 2yz) dy + (2x= − y²) dz for every piecewise-smooth curve C (i) from (1, 0, 1) to (3, 2, -1)? (ii) from (3, 2, -1) to (1, 0, 1)?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 12E
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![Let h be the vector field
h(x, y, z) = (2xy +=²)i + (x² − 2yz)j + (2x= − y²) k.
-
(a) Show that h is a gradient field.
(b) What is the value of
[((2xy +2²) dx + (x² − 2yz) dy + (2x= − y²) dz
for every piecewise-smooth curve C (i) from (1, 0, 1) to
(3, 2, -1)? (ii) from (3, 2, -1) to (1, 0, 1)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F19383c96-ec1e-42f4-a384-7094e148b655%2Fa1b8a40e-b482-4b31-ad3b-031c0ef3debb%2Fhgg8uy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let h be the vector field
h(x, y, z) = (2xy +=²)i + (x² − 2yz)j + (2x= − y²) k.
-
(a) Show that h is a gradient field.
(b) What is the value of
[((2xy +2²) dx + (x² − 2yz) dy + (2x= − y²) dz
for every piecewise-smooth curve C (i) from (1, 0, 1) to
(3, 2, -1)? (ii) from (3, 2, -1) to (1, 0, 1)?
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