The Fundamental Theorem for Line Integrals / The Gradient Theorem 1. Consider the scalar field f(x,y, z) = 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, t) Osts1 H: r(t) = (cost,sin t, 2t/m) 0stSa/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: · 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) Compute ,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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XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
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Can I please get help with g) h) and I)
![The Fundamental Theorem for Line Integrals / The Gradient Theorem
1. Consider the scalar field
f(x,y,z) = 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:
r(t) = (1- t, t, t) 0sts1
H: r(t) = (cost,sin t , 2t/m) 0<t<n/2
L:
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).
g) Compute f, F · dr.
h) Computes, 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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F430be2f4-e615-4e98-bd4a-b13b5606f96a%2Fc7ea38bc-a978-4b80-b288-8e6e9936f78e%2Fedz0a6_processed.jpeg&w=3840&q=75)
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