Q. 14 Write 鈭獵F(x,y,z)路dr explicitl... [FREE SOLUTION]

Chapter 14: Q. 14 (page 1096)

Write ${鈭�}_{C} F (x, y, z) 路 d r$ explicitly as an integral of t, where $F (x, y, z) = (z y 鈭� x, x z 鈭� y, x y 鈭� z)$ and role="math" localid="1650820575074" $r (t) = (t, \cos t, \sin t)$ for $a 鈮� t 鈮� b$ .

Short Answer

Expert verified

Hence, ${鈭�}_{C} F (x, y, z) 路 d r = {鈭�}_{a}^{b} \sin t \cos t - t - t \cos 2 t d t$

Step by step solution

Step 1. Given Information

Write ${鈭�}_{C} F (x, y, z) 路 d r$ explicitly as an integral of t, where role="math" localid="1650822686518" $F (x, y, z) = (z y 鈭� x, x z 鈭� y, x y 鈭� z)$ and role="math" localid="1650821798839" $r (t) = (t, \cos t, \sin t)$ for $a 鈮� t 鈮� b$ .

Step 2. The curve parameterize by r(t)=(t, cost, sint) at a≤t≤b

Thus $r' (t) = (1, - \sin t, \cos t)$ . So,

${鈭�}_{C} f (x, y, z) dr = {鈭�}_{a}^{b} f (r (t) 路 r' (t) d t {鈭�}_{C} F (x, y, z) 路 dr = {鈭�}_{a}^{b} (\sin t \cos t - t, t \sin t - \cos t, t \cos t - \sin t) (1, - \sin t, \cos t) d t {鈭�}_{C} F (x, y, z) 路 d r = {鈭�}_{a}^{b} 1 (\sin t \cos t - t) - \sin t (t \sin t - \cos t) + \cos t (t \cos t - \sin t) d t {鈭�}_{C} F (x, y, z) 路 d r = {鈭�}_{a}^{b} \sin t \cos t - t - t \sin^{2} t + \sin t \cos t + t \cos^{2} t - \sin t \cos t d t {鈭�}_{C} F (x, y, z) 路 d r = {鈭�}_{a}^{b} \sin t \cos t - t + t (\cos^{2} t - \sin^{2} t) d t {鈭�}_{C} F (x, y, z) 路 d r = {鈭�}_{a}^{b} \sin t \cos t - t - t \cos 2 t d t$

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91影视

Write ${鈭�}_{C} F (x, y, z) 路 d r$ explicitly as an integral of t, where $F (x, y, z) = (z y 鈭� x, x z 鈭� y, x y 鈭� z)$ and role="math" localid="1650820575074" $r (t) = (t, \cos t, \sin t)$ for $a 鈮� t 鈮� b$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The curve parameterize by r(t)=(t, cost, sint) at a≤t≤b

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91影视

Write 鈭�CF(x,y,z)路dr explicitly as an integral of t, where F(x,y,z)=(zy鈭�x,xz鈭�y,xy鈭�z) and role="math" localid="1650820575074" r(t)=(t,cost,sint) for a鈮�t鈮�b.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The curve parameterize by r(t)=(t, cost, sint) at a≤t≤b

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Pure Maths

Decision Maths

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Write ${鈭�}_{C} F (x, y, z) 路 d r$ explicitly as an integral of t, where $F (x, y, z) = (z y 鈭� x, x z 鈭� y, x y 鈭� z)$ and role="math" localid="1650820575074" $r (t) = (t, \cos t, \sin t)$ for $a 鈮� t 鈮� b$ .