3. (i) Using the definition of the line integral of a vector field, calculate the line integral LF. F.dy of the vector field F: R2 R2 given by F(x, y) = (y, x), and where the curve is the unit semi-circle centred at the origin, located in the upper half-plane and oriented in the anticlockwise direction. Hint. Represent the curve y as the join of two curves 7 = 71 + 1/2 (see Example 8.9 in the Notes). [20 Marks] (ii) Calculate the same integral using Green's Theorem. [10 Marks]
3. (i) Using the definition of the line integral of a vector field, calculate the line integral LF. F.dy of the vector field F: R2 R2 given by F(x, y) = (y, x), and where the curve is the unit semi-circle centred at the origin, located in the upper half-plane and oriented in the anticlockwise direction. Hint. Represent the curve y as the join of two curves 7 = 71 + 1/2 (see Example 8.9 in the Notes). [20 Marks] (ii) Calculate the same integral using Green's Theorem. [10 Marks]
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![3. (i) Using the definition of the line integral of a vector field, calculate the
line integral
LF.
F.dy
of the vector field F: R2 R2 given by
F(x, y) = (y, x),
and where the curve is the unit semi-circle centred at the origin, located in
the upper half-plane and oriented in the anticlockwise direction.
Hint. Represent the curve y as the join of two curves 7 = 71 + 1/2 (see Example 8.9
in the Notes).
[20 Marks]
(ii) Calculate the same integral using Green's Theorem.
[10 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c7573-6777-48ff-9bfb-9b3df1a769a6%2Fe2d22b54-7813-4c83-963b-e86a376c6874%2Fbrv3oo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. (i) Using the definition of the line integral of a vector field, calculate the
line integral
LF.
F.dy
of the vector field F: R2 R2 given by
F(x, y) = (y, x),
and where the curve is the unit semi-circle centred at the origin, located in
the upper half-plane and oriented in the anticlockwise direction.
Hint. Represent the curve y as the join of two curves 7 = 71 + 1/2 (see Example 8.9
in the Notes).
[20 Marks]
(ii) Calculate the same integral using Green's Theorem.
[10 Marks]
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