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Calculate the Green’s value for the functions F = y2 and G = x2 for the region x = 1 and y = 2 from origin.
- a)0
- b)2
- c)-2
- d)1
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Calculate the Green’s value for the functions F = y2and G = x2fo...
Answer: c
Explanation: ∫∫(dG/dx – dF/dy)dx dy = ∫∫(2x – 2y)dx dy. On integrating for x = 0->1 and y = 0->2, we get Green’s value as -2.
Explanation: ∫∫(dG/dx – dF/dy)dx dy = ∫∫(2x – 2y)dx dy. On integrating for x = 0->1 and y = 0->2, we get Green’s value as -2.
Most Upvoted Answer
Calculate the Green’s value for the functions F = y2and G = x2fo...
Calculating Green's value for the given functions
Green's value for a region can be calculated by finding the double integral of the curl of the given functions over the region. In this case, the functions F = y^2 and G = x^2 are given for the region x = 1 and y = 2.
Calculating the curl of the given functions
The curl of a vector field F = P(x, y)i + Q(x, y)j is given by the expression:
curl(F) = (∂Q/∂x - ∂P/∂y)k
For the given functions F = y^2 and G = x^2, we have:
P(x, y) = y^2 and Q(x, y) = x^2
Calculating the partial derivatives:
∂P/∂y = 2y
∂Q/∂x = 2x
Calculating the curl:
curl(F) = (2x - 2y)k
Calculating the double integral over the region
The double integral of the curl over the region x = 1 and y = 2 can be calculated as follows:
∫∫(curl(F)) dA = ∫∫(2x - 2y) dA
Integrating with respect to x and then y:
∫(2x - 2y)dx = x^2 - 2xy
∫(x^2 - 2xy)dy = x^2y - xy^2
Substitute the limits x = 1 and y = 2 into the expression:
x^2y - xy^2 = 1*2 - 1*2 = -2
Therefore, the Green's value for the given functions F = y^2 and G = x^2 for the region x = 1 and y = 2 from the origin is -2, which corresponds to option 'C'.
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