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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)?

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)?

Advanced Engineering Mathematics
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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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)?
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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