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Find the value of Stoke’s theorem for A = x i + y j + z k. The state of the function will be
- a)Solenoidal
- b)Divergent
- c)Rotational
- d)Curl free
Correct answer is option 'D'. Can you explain this answer?
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Find the value of Stoke’s theorem for A = x i + y j + z k. The s...
Understanding Stoke's Theorem
Stoke's Theorem relates the surface integral of a vector field over a surface to the line integral of the field around the boundary of that surface. For a vector field A, it can be expressed as:
∮C A · dr = ∬S (curl A) · dS
Where C is the boundary curve, S is the surface, and curl A represents the curl of vector field A.
Analyzing the Vector Field A
Given the vector field A = x i + y j + z k:
- Components:
- A has three components: x, y, z, which correspond to the x, y, and z directions, respectively.
Calculating the Curl of A
To determine if A is curl-free (or rotational), we calculate the curl:
- Curl A = ∇ × A:
- For A = x i + y j + z k, the curl results in:
- curl A = (∂z/∂y - ∂y/∂z) i + (∂x/∂z - ∂z/∂x) j + (∂y/∂x - ∂x/∂y) k
- All partial derivatives are zero, leading to:
curl A = 0 i + 0 j + 0 k = 0
Conclusion: Curl Free
Since curl A = 0, it indicates that the vector field A is:
- Curl Free: No rotation or circulation around any point in space.
- This aligns with option 'D' provided in the question.
Understanding the Correct Answer
Thus, the correct answer is indeed option 'D' - the vector field A is curl-free, meaning it has no rotational component. This is a crucial concept in fluid dynamics and electromagnetism, where curl-free fields often correspond to conservative forces.
Stoke's Theorem relates the surface integral of a vector field over a surface to the line integral of the field around the boundary of that surface. For a vector field A, it can be expressed as:
∮C A · dr = ∬S (curl A) · dS
Where C is the boundary curve, S is the surface, and curl A represents the curl of vector field A.
Analyzing the Vector Field A
Given the vector field A = x i + y j + z k:
- Components:
- A has three components: x, y, z, which correspond to the x, y, and z directions, respectively.
Calculating the Curl of A
To determine if A is curl-free (or rotational), we calculate the curl:
- Curl A = ∇ × A:
- For A = x i + y j + z k, the curl results in:
- curl A = (∂z/∂y - ∂y/∂z) i + (∂x/∂z - ∂z/∂x) j + (∂y/∂x - ∂x/∂y) k
- All partial derivatives are zero, leading to:
curl A = 0 i + 0 j + 0 k = 0
Conclusion: Curl Free
Since curl A = 0, it indicates that the vector field A is:
- Curl Free: No rotation or circulation around any point in space.
- This aligns with option 'D' provided in the question.
Understanding the Correct Answer
Thus, the correct answer is indeed option 'D' - the vector field A is curl-free, meaning it has no rotational component. This is a crucial concept in fluid dynamics and electromagnetism, where curl-free fields often correspond to conservative forces.
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Find the value of Stoke’s theorem for A = x i + y j + z k. The state of the function will bea)Solenoidalb)Divergentc)Rotationald)Curl freeCorrect answer is option 'D'. Can you explain this answer?
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