GATE EC 2022 Set 1 — Question 11
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Engineering Mathematics → Vector Calculus → Line, Surface & Volume Integrals
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Question
Consider the two-dimensional vector field , where and denote the unit vectors along the x-axis and the y-axis, respectively. A contour in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral is
Correct answer
(A) 0
Solution
The given vector field is .
We need to evaluate the closed path integral .
This integral can be written as .We can use Green's Theorem, which states that for a vector field , the line integral over a closed curve is given by:
In our case, and .
So, .
And .Therefore, .Applying Green's Theorem:
.Alternatively, we can recognize that is a conservative vector field. A vector field is conservative if it is the gradient of some scalar potential function , i.e., .
For , we can find . Then and .
For a conservative vector field, the line integral over any closed path is zero.Thus, the value of the closed path integral is 0.The final answer is
We need to evaluate the closed path integral .
This integral can be written as .We can use Green's Theorem, which states that for a vector field , the line integral over a closed curve is given by:
In our case, and .
So, .
And .Therefore, .Applying Green's Theorem:
.Alternatively, we can recognize that is a conservative vector field. A vector field is conservative if it is the gradient of some scalar potential function , i.e., .
For , we can find . Then and .
For a conservative vector field, the line integral over any closed path is zero.Thus, the value of the closed path integral is 0.The final answer is
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