3. (i) Using the definition of the line integral of a vector field, calculate theline integralLF.F.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 7 = 71 + 1/2 (see Example 8.9in the Notes).[20 Marks](ii) Calculate the same integral using Green's Theorem.[10 Marks]

Step-by-Step Solution We are tasked with calculating the line integral of the vector field…

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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Use the equation giving the flux of the vector field across the curve to calculate the flux of x + 1 y lã (x + 1)² + y²' (x + 1)² + y² F(x, y) = across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) I F. dr =
R.p Plot the gradient vector field of f together with a contour map of f. Explain how they are related to each other. f (x, y) = ln(1 + x^2 + 2y^2)
calculate div(F) and curl(F). F = (xy, yz, y² – x³)
For vector field F = (-2y, y), y > 0. Find all points P such that the amount of fluid flowing into P equals the amount of fluid flowing out of P. Write down the equation these points satisfy.
3. Let F and G be vector ficlds with differentiable components. Express curl (F x G) in term of div and dot products.
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.
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 slope field for the points (ti, y;) where at and dy df y-2 Z₁ = -1 +0.25 (1-1) = for 121,2,5,... 9 yj Y; = 1 +0.25 (j-1) for j=1,2,3,... 9
Show all steps please
Use the equation giving the flux of the vector field across the curve to calculate the flux of x+1 y (₁. (x + 1)² + y²' (x + 1)² + y² F(x, y): across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) [F C = Incorrect F. dr = 0.1074

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