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Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The relation between the area elements dx'dy' and dx'dy' is given by dx'dy'=jdxdy. The value of j is (a)2 (b)1 (c) -1 (d)-2 The correct option is c please anyone solve this?
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Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The re...
Coordinates Transformation
The given coordinates transformation is x' = (x + y) / √2 and y' = (x - y) / √2. This means that the new coordinates (x', y') are expressed in terms of the old coordinates (x, y).
Calculating the Area Elements
To find the relation between the area elements dx'dy' and dxdy, we can use the Jacobian determinant. The Jacobian determinant is given by:
J = (∂(x', y') / ∂(x, y)) = |∂x' / ∂x ∂x' / ∂y|
|∂y' / ∂x ∂y' / ∂y|
Taking the partial derivatives of x' and y' with respect to x and y, we get:
∂x' / ∂x = 1/√2
∂x' / ∂y = 1/√2
∂y' / ∂x = 1/√2
∂y' / ∂y = -1/√2
Substituting these values into the Jacobian determinant formula, we have:
J = |1/√2 1/√2|
|1/√2 -1/√2|
Calculating the Determinant
Calculating the determinant of this matrix, we have:
J = (1/√2)(-1/√2) - (1/√2)(1/√2)
= -1/2 - 1/2
= -1
Therefore, the value of j is -1.
Answer
The correct option is (c) -1.
The given coordinates transformation is x' = (x + y) / √2 and y' = (x - y) / √2. This means that the new coordinates (x', y') are expressed in terms of the old coordinates (x, y).
Calculating the Area Elements
To find the relation between the area elements dx'dy' and dxdy, we can use the Jacobian determinant. The Jacobian determinant is given by:
J = (∂(x', y') / ∂(x, y)) = |∂x' / ∂x ∂x' / ∂y|
|∂y' / ∂x ∂y' / ∂y|
Taking the partial derivatives of x' and y' with respect to x and y, we get:
∂x' / ∂x = 1/√2
∂x' / ∂y = 1/√2
∂y' / ∂x = 1/√2
∂y' / ∂y = -1/√2
Substituting these values into the Jacobian determinant formula, we have:
J = |1/√2 1/√2|
|1/√2 -1/√2|
Calculating the Determinant
Calculating the determinant of this matrix, we have:
J = (1/√2)(-1/√2) - (1/√2)(1/√2)
= -1/2 - 1/2
= -1
Therefore, the value of j is -1.
Answer
The correct option is (c) -1.
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Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The relation between the area elements dx'dy' and dx'dy' is given by dx'dy'=jdxdy. The value of j is (a)2 (b)1 (c) -1 (d)-2 The correct option is c please anyone solve this?
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Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The relation between the area elements dx'dy' and dx'dy' is given by dx'dy'=jdxdy. The value of j is (a)2 (b)1 (c) -1 (d)-2 The correct option is c please anyone solve this? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The relation between the area elements dx'dy' and dx'dy' is given by dx'dy'=jdxdy. The value of j is (a)2 (b)1 (c) -1 (d)-2 The correct option is c please anyone solve this? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The relation between the area elements dx'dy' and dx'dy' is given by dx'dy'=jdxdy. The value of j is (a)2 (b)1 (c) -1 (d)-2 The correct option is c please anyone solve this?.
Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The relation between the area elements dx'dy' and dx'dy' is given by dx'dy'=jdxdy. The value of j is (a)2 (b)1 (c) -1 (d)-2 The correct option is c please anyone solve this? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The relation between the area elements dx'dy' and dx'dy' is given by dx'dy'=jdxdy. The value of j is (a)2 (b)1 (c) -1 (d)-2 The correct option is c please anyone solve this? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the coordinates transformation x'=(x y)/√2,y'=(x-y)/√2.The relation between the area elements dx'dy' and dx'dy' is given by dx'dy'=jdxdy. The value of j is (a)2 (b)1 (c) -1 (d)-2 The correct option is c please anyone solve this?.
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