3. (i) Using the definition of the line integral of a vector field, calculate theline integralL³F.dyof the vector field F: R² → R² given byF(x, y) = (y, x),and where the curve & is the unit semi-circle centred at the origin, located inthe upper half-plane and oriented in the anticlockwise direction.Hint. Represent the curve y as the join of two curves y = 71 + 1/2 (see Example 8.9in the Notes).[20 Marks](ii) Calculate the same integral using Green's Theorem.[10 Marks]

Answered: Image /qna-images/answer/bd17c68c-9fa3-4240-80e7-77a2a630e1ea.jpg

Answered: 3. (i) Using the definition of the line…

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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A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x - y, z + y + 9, z?) and the net is decribed by the equation y = V1 - x² – z7, y > 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.) v · dS = 10n Incorrect
answer in fraction please thanks
A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x - y, z + y + 6, z? ) and the net is decribed by the equation y = 1 – x - z', y 2 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.) · dS = Incorrect
1. Let F = -yi + xj and let C be the unit circle oriented counterclockwise. (a) Show that F has a constant magnitude of 1 on C. (b) Show that F is always tangent to the circle C. (c) Show that F - dr =length of C. (d) Without calculating the curl, determine whether F is path-independent. Explain your reasoning. 2. Consider the vector field F(x, y) = evi+ (xe" + e²)j + ye*k. %3D (a) Show that F is conservative by determining a function f such that F = Vf. (b) Find a parameterization of C, the line segment from (0, 2, 0) to (4, 0, 3). (c) Set up the line integral f. F - dr, where C is the line segment from (0, 2, 0) to (4,0, 3). (d) Evaluate fe F - dr using whatever method you think is best. 3. Let i = V(r? + y?). Consider the path C which is a line between any two of the following points: (0,0), (-5,0), (0, 6), (0, –6), (5,4), (-3, –5). Suppose you want to choose the path C in order to maximize ở - dr. What point should be the start of C? What point should be the end of C? Explain your…
Sketch the vector ﬁeld in the xy-plane and find the work... along the arc of a unit circle from (1, 0) to (0, 1)
Evaluate This Integral if curve C consists of curve C₁ which is a parabola y=x² from point (0,0) to point (2,4) and curve C₂ which is a vertical line segment from point (2,4) to point (2,6) if a and b are each constant.
Let us verify Stokes' theorem using the vector field F = (x 2 - y)i + 4zj + x 2k, where the closed contour consists of the x and y coordinate axes and that portion of the circle 2 + y 2 = a 2 that lies in the first quadrant with z = 1
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Consider the triangle shown in the figure below. a. Find the equation of the straight edge OA. b. Find the equation of the straight edge AB.

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