Please answer these two questions using the following example: Parameterize the given C in two different ways to evaluat
Posted: Wed Jul 06, 2022 12:33 pm
Please answer these two questions using the followingexample:
Parameterize the given C in two different ways to evaluate theline integral.
Verify that the line integral does NOT depend on the choice ofparameterization.
Thank you!
Example 6 Solution Consider the oriented path which is a straight-line segment L running from (0, 0) to (1, 1). Calculate the line integral of the vector field F = (3x − y)ỉ + xj along L using each of the parameterizations (a) A(t) = (t, t), 0≤t≤ 1, (b) D(t) = (e¹ - 1, e¹-1), 0≤ t ≤ In 2. The line L has equation y = x. Both A(t) and D(t) give a parameterization of L: each has both coordinates equal and each begins at (0,0) and ends at (1,1). Now let's calculate the line integral of the vector field F = (3x − y)i + xj using each parameterization. (a) Using A(t), we get li (b) Using D(t), we get F.dr = 1,² F.dr = ['((3 '((31 − 1)Ï + 1] ) · (ï + 7) dt = ["' 31 3t dt = - 3/2/1/1 = ² In 2 √¨´ ((3(e¹ − 1) − (e¹ − 1))i + (e' − 1)¡ ) · (e'ï + e'j') dt - In 2 In 2 3 = √3(e²¹ - e¹) dt = 3 ³ ( ²2² 2 - 0 ) | D ² = ²2.
(2) As discussed above, the parameterization for C is not unique. The result of NOT depend on the parameterization chosen. (Text 18.2 / 34) [7.d7 does
Parameterize the given C in two different ways to evaluate theline integral.
Verify that the line integral does NOT depend on the choice ofparameterization.
- Please Answer These Two Questions Using The Following Example Parameterize The Given C In Two Different Ways To Evaluat 1 (292.46 KiB) Viewed 13 times
Example 6 Solution Consider the oriented path which is a straight-line segment L running from (0, 0) to (1, 1). Calculate the line integral of the vector field F = (3x − y)ỉ + xj along L using each of the parameterizations (a) A(t) = (t, t), 0≤t≤ 1, (b) D(t) = (e¹ - 1, e¹-1), 0≤ t ≤ In 2. The line L has equation y = x. Both A(t) and D(t) give a parameterization of L: each has both coordinates equal and each begins at (0,0) and ends at (1,1). Now let's calculate the line integral of the vector field F = (3x − y)i + xj using each parameterization. (a) Using A(t), we get li (b) Using D(t), we get F.dr = 1,² F.dr = ['((3 '((31 − 1)Ï + 1] ) · (ï + 7) dt = ["' 31 3t dt = - 3/2/1/1 = ² In 2 √¨´ ((3(e¹ − 1) − (e¹ − 1))i + (e' − 1)¡ ) · (e'ï + e'j') dt - In 2 In 2 3 = √3(e²¹ - e¹) dt = 3 ³ ( ²2² 2 - 0 ) | D ² = ²2.
(2) As discussed above, the parameterization for C is not unique. The result of NOT depend on the parameterization chosen. (Text 18.2 / 34) [7.d7 does