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Answer: c
Explanation: The Green’s theorem states that if L and M are functions of (x,y) in an open region containing D and having continuous partial derivatives then,
∫ (F dx + G dy) = ∫∫(dG/dx – dF/dy)dx dy, with path taken anticlockwise.
Find the value of Green’s theorem for F = x^{2} and G = y^{2} is
Answer: a
Explanation: ∫∫(dG/dx – dF/dy)dx dy = ∫∫(0 – 0)dx dy = 0. The value of Green’s theorem gives zero for the functions given.
Which of the following is not an application of Green’s theorem?
Answer: c
Explanation: In physics, Green’s theorem is used to find the two dimensional flow integrals. In plane geometry, it is used to find the area and centroid of plane figures.
Answer: b
Explanation: The Green’s theorem calculates the area traversed by the functions in the region in the anticlockwise direction. This converts the line integral to surface integral.
Calculate the Green’s value for the functions F = y^{2} and G = x^{2} for the region x = 1 and y = 2 from origin.
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.
If two functions A and B are discrete, their Green’s value for a region of circle of radius a in the positive quadrant is
Answer: d
Explanation: Green’s theorem is valid only for continuous functions. Since the given functions are discrete, the theorem is invalid or does not exist.
Answer: b
Explanation: Since Green’s theorem converts line integral to surface integral, we get the value as two dimensional. In other words the functions are variable with respect to x,y, which is two dimensional.
The Green’s theorem can be related to which of the following theorems mathematically?
Answer: b
Explanation: The Green’s theorem is a special case of the Kelvin Stokes theorem, when applied to a region in the xy plane. It is a widely used theorem in mathematics and physics.
he Shoelace formula is a shortcut for the Green’s theorem. State True/False.
Answer: a
Explanation: The Shoelace theorem is used to find the area of polygon using cross multiples. This can be verified by dividing the polygon into triangles. It is a special case of Green’s theorem.
Find the area of a right angled triangle with sides of 90 degree unit and the functions described by L = cos y and M = sin x.
Answer: d
Explanation: dM/dx = cos x and dL/dy = sin y
∫∫(dM/dx – dL/dy)dx dy = ∫∫ (cos x + sin y)dx dy. On integrating with x = 0>90 and y = 0>90, we get area of right angled triangle as 180 units (taken in clockwise direction). Since area cannot be negative, we take 180 units.
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