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Compute the divergence of the vector xi + yj + zk.
- a)0
- b)1
- c)2
- d)3
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Compute the divergence of the vector xi + yj + zk.a)0b)1c)2d)3Correct ...
Answer: d
Explanation: The vector given is a position vector. The divergence of any position vector is always 3.
Explanation: The vector given is a position vector. The divergence of any position vector is always 3.
Most Upvoted Answer
Compute the divergence of the vector xi + yj + zk.a)0b)1c)2d)3Correct ...
Understanding Divergence
Divergence is a mathematical operator applied to a vector field that measures the magnitude of a source or sink at a given point. It is calculated using the following formula:
Divergence of a vector field F = ∇ · F
Where F = xi + yj + zk. Here, F is expressed in Cartesian coordinates.
Step-by-Step Calculation
1. Components of the Vector Field:
- F = xi + yj + zk
- Components are:
- F1 = x
- F2 = y
- F3 = z
2. Applying the Divergence Formula:
- Divergence is calculated as:
- ∇ · F = (∂F1/∂x) + (∂F2/∂y) + (∂F3/∂z)
3. Calculating Each Partial Derivative:
- ∂F1/∂x = ∂(x)/∂x = 1
- ∂F2/∂y = ∂(y)/∂y = 1
- ∂F3/∂z = ∂(z)/∂z = 1
4. Summing the Partial Derivatives:
- ∇ · F = 1 + 1 + 1 = 3
Conclusion
Thus, the divergence of the vector field xi + yj + zk is 3.
Final Answer:
The correct option is 'D' (3).
This result indicates that the vector field has a uniform source density throughout the space.
Divergence is a mathematical operator applied to a vector field that measures the magnitude of a source or sink at a given point. It is calculated using the following formula:
Divergence of a vector field F = ∇ · F
Where F = xi + yj + zk. Here, F is expressed in Cartesian coordinates.
Step-by-Step Calculation
1. Components of the Vector Field:
- F = xi + yj + zk
- Components are:
- F1 = x
- F2 = y
- F3 = z
2. Applying the Divergence Formula:
- Divergence is calculated as:
- ∇ · F = (∂F1/∂x) + (∂F2/∂y) + (∂F3/∂z)
3. Calculating Each Partial Derivative:
- ∂F1/∂x = ∂(x)/∂x = 1
- ∂F2/∂y = ∂(y)/∂y = 1
- ∂F3/∂z = ∂(z)/∂z = 1
4. Summing the Partial Derivatives:
- ∇ · F = 1 + 1 + 1 = 3
Conclusion
Thus, the divergence of the vector field xi + yj + zk is 3.
Final Answer:
The correct option is 'D' (3).
This result indicates that the vector field has a uniform source density throughout the space.
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