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A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is
- a)4xy − 2x2 + 2y2 + constant
- b)4y2 − 4xy + constant
- c)2x2 − 2y2 + xy + constant
- d)−4xy + 2y2 − 2x2 + constant
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A harmonic function is analytic if it satisfies the Laplace equation. ...
To determine if the function u(x, y) = 2x^2 satisfies the Laplace equation and is therefore analytic, we need to calculate its Laplacian.
The Laplacian of a function u(x, y) is given by the second partial derivatives with respect to x and y:
∇²u = ∂²u/∂x² + ∂²u/∂y²
Let's calculate the partial derivatives of u(x, y):
∂u/∂x = 4x
∂²u/∂x² = 4
∂u/∂y = 0
∂²u/∂y² = 0
Now, let's calculate the Laplacian:
∇²u = ∂²u/∂x² + ∂²u/∂y² = 4 + 0 = 4
Since the Laplacian of u(x, y) = 2x^2 is equal to 4 and not zero, it does not satisfy the Laplace equation. Therefore, u(x, y) = 2x^2 is not analytic.
The Laplacian of a function u(x, y) is given by the second partial derivatives with respect to x and y:
∇²u = ∂²u/∂x² + ∂²u/∂y²
Let's calculate the partial derivatives of u(x, y):
∂u/∂x = 4x
∂²u/∂x² = 4
∂u/∂y = 0
∂²u/∂y² = 0
Now, let's calculate the Laplacian:
∇²u = ∂²u/∂x² + ∂²u/∂y² = 4 + 0 = 4
Since the Laplacian of u(x, y) = 2x^2 is equal to 4 and not zero, it does not satisfy the Laplace equation. Therefore, u(x, y) = 2x^2 is not analytic.
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Community Answer
A harmonic function is analytic if it satisfies the Laplace equation. ...
Concept:
Let w = u + iν be a function of complex variable.
Function of a complex variable is analytic, if it satisfies Cauchy-reimann equation;
Calculation:
Given:
Let w = u + iν be a function of complex variable.
Function of a complex variable is analytic, if it satisfies Cauchy-reimann equation;
Calculation:
Given:
u(x, y) = 2x2 – 2y2 + 4xy, ν(x, y) = ?
∂u/∂x = ∂ν/∂y
∂u/∂x = 4x + 4y = ∂ν/∂y
∂u/∂x = 4x + 4y = ∂ν/∂y
Integrating w.r.t y keeping x constant
ν(x, y) = 4xy + 2y2 + f(x)
∂v/∂x = 4y + f′(x)
∂u/∂y =−∂ν*∂x
∂u/∂y = −4y+4x
4y – 4x = 4y + f’(x)
f(x) = −4x2*2 + C = −2x2 + C
∴ ν(x, y) = 4xy + 2y2 – 2x2 + C
∂v/∂x = 4y + f′(x)
∂u/∂y =−∂ν*∂x
∂u/∂y = −4y+4x
4y – 4x = 4y + f’(x)
f(x) = −4x2*2 + C = −2x2 + C
∴ ν(x, y) = 4xy + 2y2 – 2x2 + C
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A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer?
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A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer?.
A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer?.
Solutions for A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Civil Engineering (CE).
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ample number of questions to practice A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice Civil Engineering (CE) tests.
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