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The expression curl (grad f), where f is a scalar function, is
- a)Equal to ∇2f
- b)Equal to div (grad f)
- c)A scalar of zero magnitude
- d)A vector of zero magnitude
Correct answer is option 'D'. Can you explain this answer?
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The expression curl (grad f), where f is a scalar function, is a)Equal...
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The expression curl (grad f), where f is a scalar function, is a vector of zero magnitude.
Explanation:
1. Curl and Gradient
To understand the expression curl (grad f), we need to first understand the concepts of curl and gradient.
- Curl: The curl of a vector field measures the rotation or circulation of the field at a particular point. It is denoted by curl(F) or ∇ × F, where ∇ is the del operator.
- Gradient: The gradient of a scalar function measures the rate of change or the slope of the function at a particular point. It is denoted by grad(f) or ∇f.
2. Curl of Gradient
Now, let's consider the expression curl (grad f), where f is a scalar function.
- First, we find the gradient of the scalar function f, which is given by grad(f) = ∇f. The gradient is a vector field.
- Next, we take the curl of the gradient, which is given by curl (grad f) or ∇ × (∇f).
3. Curl of Gradient of a Scalar Function
For any scalar function f, the curl of its gradient is always a vector of zero magnitude. This means that the curl (grad f) = 0.
- The reason for this is that the curl measures the rotation or circulation of a vector field. However, the gradient of a scalar function is a conservative vector field, which means it has zero curl.
- In other words, the gradient of a scalar function represents a field with no rotation or circulation. Therefore, taking the curl of such a field will always result in zero.
4. Conclusion
In conclusion, the expression curl (grad f), where f is a scalar function, is a vector of zero magnitude. This is because the gradient of a scalar function represents a field with no rotation or circulation, and taking the curl of such a field always results in zero.
Explanation:
1. Curl and Gradient
To understand the expression curl (grad f), we need to first understand the concepts of curl and gradient.
- Curl: The curl of a vector field measures the rotation or circulation of the field at a particular point. It is denoted by curl(F) or ∇ × F, where ∇ is the del operator.
- Gradient: The gradient of a scalar function measures the rate of change or the slope of the function at a particular point. It is denoted by grad(f) or ∇f.
2. Curl of Gradient
Now, let's consider the expression curl (grad f), where f is a scalar function.
- First, we find the gradient of the scalar function f, which is given by grad(f) = ∇f. The gradient is a vector field.
- Next, we take the curl of the gradient, which is given by curl (grad f) or ∇ × (∇f).
3. Curl of Gradient of a Scalar Function
For any scalar function f, the curl of its gradient is always a vector of zero magnitude. This means that the curl (grad f) = 0.
- The reason for this is that the curl measures the rotation or circulation of a vector field. However, the gradient of a scalar function is a conservative vector field, which means it has zero curl.
- In other words, the gradient of a scalar function represents a field with no rotation or circulation. Therefore, taking the curl of such a field will always result in zero.
4. Conclusion
In conclusion, the expression curl (grad f), where f is a scalar function, is a vector of zero magnitude. This is because the gradient of a scalar function represents a field with no rotation or circulation, and taking the curl of such a field always results in zero.
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The expression curl (grad f), where f is a scalar function, is a)Equal to ∇2f b)Equal to div (grad f) c)A scalar of zero magnitude d)A vector of zero magnitude Correct answer is option 'D'. Can you explain this answer?
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