Exams › GATE › Engineering Mathematics
Let R be a region in the first quadrant of the xy plane enclosed by a closed curve C considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region R?
- ∬_R dxdy
- ∮_C xdy
- ∮_C ydx
- 1/2 ∮_C (xdy − ydx)
Correct answer: ∮_C ydx
Solution
The expression ∮_C ydx represents the circulation of the vector field (0, y) around the curve C, which does not calculate the area of region R. In contrast, the other options either directly compute the area or relate to it through Green's Theorem.
Related GATE Engineering Mathematics questions
- A vector field p and a scalar field r are given by
p = (2x² - 3xy + z²) î + (2y² - 3yz + x²) ĵ + (2z² - 3xz + x²) k̂
r = 6x² + 4y² - z² - 9xyz - 2xy + 3xz - yz
Consider the statements P and Q.
P: Curl of the gradient of the scalar field r is a null vector.
Q: Divergence of curl of the vector field p is zero.
Which one of the following options is CORRECT?
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- A delivery agent is at a location R. To deliver the order, she is instructed to travel to location P along straight-line paths of RC, CA, AB and BP of 5 km each. The direction of each path is given in the table below as whole circle bearings. Assume that the latitude (L) and departure (D) of R is (0, 0) km. What is the latitude and departure of P (in km, rounded off to one decimal place)? Paths: RC, CA, AB, BP. Directions (in degrees): 120, 0, 90, 240.
- Three vectors p, q and r are given as p = î + ĵ + k̂, q = î + 2ĵ + 3k̂, r = 2î + 3ĵ + 4k̂. Which of the following is/are CORRECT?
- If a vector field V is related to another vector field A through V = ∇ × A, which of the following is true? Note: C and S_c refer to any closed contour and any surface whose boundary is C.
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