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The Fundamental Theorem for Line Integrals / The Gradient Theorem 1. Consider the scalar field f(x,y,2) = 3xy + z² and the two points A = (1,0,0) and B = (0,1,1). Define two different curves that start at A and end at B: L: r(t) = (1– t,t, ) Osts1 H: r(t) = (cost,sin t,2t/m) 0stsn/2 Note that L is a line segment, while H is (part of) a helix. a) First verify that both curves truly do begin at A and end at B. That is, plug in to prove that r(0) = A and r(1) = B for L, and r(0) = A and r(m/2) = B for H. b) Graph L, H, A and B using Geogebra, some other program, or by hand, to get a feel for what's going on. c) Find Vf, the gradient of f. You will use this below.

The Fundamental Theorem for Line Integrals / The Gradient Theorem 1. Consider the scalar field f(x,y,2) = 3xy + z² and the two points A = (1,0,0) and B = (0,1,1). Define two different curves that start at A and end at B: L: r(t) = (1– t,t, ) Osts1 H: r(t) = (cost,sin t,2t/m) 0stsn/2 Note that L is a line segment, while H is (part of) a helix. a) First verify that both curves truly do begin at A and end at B. That is, plug in to prove that r(0) = A and r(1) = B for L, and r(0) = A and r(m/2) = B for H. b) Graph L, H, A and B using Geogebra, some other program, or by hand, to get a feel for what's going on. c) Find Vf, the gradient of f. You will use this below.

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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The Fundamental Theorem for Line Integrals / The Gradient Theorem 1. Consider the scalar field f(x,y, z) = 3xy + z? and the two points A = (1,0,0) and B = (0,1,1). Define two different curves that start at A and end at B: L: r(t) = (1 - t, t,t) 0sts1 H: r(t) = (cos t, sint, 2t/m) 0stsn/2 Note that L is a line segment, while H is (part of) a helix. a) First verify that both curves truly do begin at A and end at B. That is, plug in to prove that r(0) = A and r(1) = B for L, and r(0) = A and r(1/2) = B for H. b) Graph L, H, A and B using Geogebra, some other program, or by hand, to get a feel for what's going on. c) Find Vf, the gradient of f. You will use this below. Now our goal is to verify the Gradient Theorem – twice! The Gradient Theorem claims that: (vf - dr = vf •dr = f(B) – f(A) d) First evaluate the leftmost expression directly, the line integral of the gradient of f along the line L. e) Next evaluate the middle expression directly, the line integral of the gradient of f along the helix H. f) Finally evaluate the rightmost expression directly, which is f(0,1,1) – f(1,0,0). This is by far the easiest part of this problem, so you're not missing something if you think it's simple. All three answers from d, e, and f should match, verifying that the Gradient Theorem is true in this case. But wait, there's more! Let F(x, y,z) = (-y,x,xy). 8) Compute S, F · dr. h) Computef, F · dr. i) The two values from parts (g) and (h) are not the same. Why not? What property of the vector field F prevents these two line integrals from being equal?
Transcribed Image Text:The Fundamental Theorem for Line Integrals / The Gradient Theorem 1. Consider the scalar field f(x,y, z) = 3xy + z? and the two points A = (1,0,0) and B = (0,1,1). Define two different curves that start at A and end at B: L: r(t) = (1 - t, t,t) 0sts1 H: r(t) = (cos t, sint, 2t/m) 0stsn/2 Note that L is a line segment, while H is (part of) a helix. a) First verify that both curves truly do begin at A and end at B. That is, plug in to prove that r(0) = A and r(1) = B for L, and r(0) = A and r(1/2) = B for H. b) Graph L, H, A and B using Geogebra, some other program, or by hand, to get a feel for what's going on. c) Find Vf, the gradient of f. You will use this below. Now our goal is to verify the Gradient Theorem – twice! The Gradient Theorem claims that: (vf - dr = vf •dr = f(B) – f(A) d) First evaluate the leftmost expression directly, the line integral of the gradient of f along the line L. e) Next evaluate the middle expression directly, the line integral of the gradient of f along the helix H. f) Finally evaluate the rightmost expression directly, which is f(0,1,1) – f(1,0,0). This is by far the easiest part of this problem, so you're not missing something if you think it's simple. All three answers from d, e, and f should match, verifying that the Gradient Theorem is true in this case. But wait, there's more! Let F(x, y,z) = (-y,x,xy). 8) Compute S, F · dr. h) Computef, F · dr. i) The two values from parts (g) and (h) are not the same. Why not? What property of the vector field F prevents these two line integrals from being equal?
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