The value of the line integral for the vector, $\overrightarrow𝑣=2𝑥̂ +𝑦𝑧 ^2𝑦̂ +(3𝑦+𝑧 ^2 )𝑧̂ $ along the closed path OABO (as shown in the figure) is: (Path AB is the arc of a circle of unit radius)

Question

The value of the line integral for the vector, $\overrightarrow𝑣=2𝑥̂ +𝑦𝑧 ^2𝑦̂ +(3𝑦+𝑧 ^2 )𝑧̂ $ along the closed path OABO (as shown in the figure) is:  (Path AB is the arc of a circle of unit radius)

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The vector field $ \vec{v} $ is given in terms of its components along the $ \hat{i} $, $ \hat{j} $, and $ \hat{k} $ directions. The path $ OABO $ forms a closed loop, and the line integral needs to be evaluated along this path. Key Observations: The path $ AB $ is a circle with a unit radius. The components of the vector field depend on $ x $, $ y $, and $ z $. The integration will involve parametric equations representing the motion of the circle in 3D space. Solution: By applying the line integral along the closed loop $ OABO $ and performing the necessary calculations, the result is: $$ \frac{1}{4} (3\pi - 1) $$ Conclusion: Thus, the correct answer is (A).

Answer

$\frac{1}{4}(3\pi-1)$

Answer

$3\pi-\frac{1}{4}$

Answer

$\frac{3\pi}{4}-1$

Answer

The value of the line integral for the vector,
\(\overrightarrow𝑣=2𝑥̂ +𝑦𝑧 ^2𝑦̂ +(3𝑦+𝑧 ^2 )𝑧̂ \)
along the closed path OABO (as shown in the figure) is:

(Path AB is the arc of a circle of unit radius)

The Correct Option is A

Solution and Explanation

Key Observations:

Solution:

Conclusion:

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The value of the line integral for the vector, \(\overrightarrow𝑣=2𝑥̂ +𝑦𝑧 ^2𝑦̂ +(3𝑦+𝑧 ^2 )𝑧̂ \)along the closed path OABO (as shown in the figure) is: (Path AB is the arc of a circle of unit radius)