Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The rotational field F = 〈–y, x〉 has zero curl and zero divergence.
b. ▿ × ▿ϕ = 0
c. Two vector fields with the same curl differ by a constant vector field.
d. Two vector fields with the same divergence differ by a constant vector field.
e. If F = 〈x, y, z) and S encloses a region D, then
a.
To explain: Whether the given statement is true or not.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given statement:
The rotational field
Calculation:
The given rotational field
Assume
The curl is
Substitute
Thus, the curl is 2.
Hence, the statement is false.
b.
To explain: Whether the given statement is true or not.
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Given statement:
The following
Calculation:
The given statement is
Here,
Hence,
It is known that the curl of a conservative field is zero.
Hence,
Therefore, the statement is true.
c.
To explain: Whether the given statement is true or not.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given statement:
Two vector fields with the same curl differ by a constant vector field.
Calculation:
Consider the rotational field
Assume
The curl is
Substitute the values
Thus, the curl is 2.
Consider the vector field
Assume
The curl is
Substitute the values
Thus, the curl is 2.
Hence, the vector fields
Thus, the statement is false.
d.
To explain: Whether the given statement is true or not.
Answer to Problem 1RE
The statement is false.
Explanation of Solution
Given statement:
Two vector fields with the same divergence differ by a constant vector field.
Calculation:
Consider the rotational field
Obtain the divergence.
Thus, the divergence is 1.
Consider the rotational field
Obtain the divergence.
Thus, the divergence is 1.
Hence, the vector fields
Thus, the statement is false.
e.
To explain: Whether the given statement is true or not.
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Given statement:
If
Theorem Used:
Divergence theorem:
Let F be a vector field whose components have continuous first partial derivatives in a connected and simply connected region D in
Calculation:
Consider the rotational field
Compute the divergence of F.
Use the Divergence theorem stated above that obtains the value of
Thus, the statement is true.
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