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F(x,y) = (x² + xy) aₓ + (y² + xy) a_y. Its line integral over the straight line from (x,y) = (0,2) to (x,y) = (2,0) evaluates to
- -8
- 4
- 8
- 0
Correct answer: 0
Solution
The line integral evaluates to zero because the vector field F is conservative, meaning it has a potential function and the integral over a path between two points depends only on the endpoints, not the path taken. Since the starting and ending points are different but the field is conservative, the integral results in zero.
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?
- What is curl of the vector field 2x²y i + 5z² j − 4yzk ?
- Let φ be a scalar field, and u be a vector field. Which of the following identities is true for div(φu)?
- 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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