Chapter 14, Problem 1RE

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 ${\displaystyle {\iint }_{S}F\cdot ndS}$ is three times the volume of D.

To determine

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 $F=\langle -y,x\rangle$ has zero curl and zero divergence.

Calculation:

The given rotational field $F=\langle -y,x\rangle$ has zero curl and zero divergence.

Assume $f\left(x,y\right)=-y$ and $g\left(x,y\right)=x$.

The curl is $\frac{\partial }{\partial x}\left(g\right)-\frac{\partial }{\partial y}\left(f\right)$.

Substitute $f\left(x,y\right)=-y$ and $g\left(x,y\right)=x$ in the curl,

$\begin{array}{c}\frac{\partial }{\partial x}\left(x\right)-\frac{\partial }{\partial y}\left(-y\right)=1-\left(-1\right)\\ =2\end{array}$

Thus, the curl is 2.

Hence, the statement is false.

To determine

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 $\nabla \times {\nabla }_{\phi }=0$ holds.

Calculation:

The given statement is $\nabla \times {\nabla }_{\phi }=0$.

Here, ${\nabla }_{\phi }$ denotes a conservative field.

Hence, $\nabla \times {\nabla }_{\phi }$ denotes the curl of the conservative field ${\nabla }_{\phi }$.

It is known that the curl of a conservative field is zero.

Hence, $\nabla \times {\nabla }_{\phi }=0$

Therefore, the statement is true.

To determine

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 $F=\langle -y,x\rangle$.

Assume $f\left(x,y\right)=-y$ and $g\left(x,y\right)=x$.

The curl is $\frac{\partial }{\partial x}\left(g\right)-\frac{\partial }{\partial y}\left(f\right)$.

Substitute the values $f\left(x,y\right)=-y$ and $g\left(x,y\right)=x$ in the formula of curl.

$\begin{array}{c}\frac{\partial }{\partial x}\left(x\right)-\frac{\partial }{\partial y}\left(-y\right)=1-\left(-1\right)\\ =2\end{array}$

Thus, the curl is 2.

Consider the vector field $F=\langle 0,2x\rangle$.

Assume $f\left(x,y\right)=0$ and $g\left(x,y\right)=2x$.

The curl is $\frac{\partial }{\partial x}\left(g\right)-\frac{\partial }{\partial y}\left(f\right)$.

Substitute the values $f\left(x,y\right)=0$ and $g\left(x,y\right)=2x$ in the formula of curl.

$\begin{array}{c}\frac{\partial }{\partial x}\left(2x\right)-\frac{\partial }{\partial y}\left(0\right)=2-0\\ =2\end{array}$

Thus, the curl is 2.

Hence, the vector fields $F=\langle -y,x\rangle$ and $F=\langle 0,2x\rangle$ has same curl but they don’t differ by a constant vector field.

Thus, the statement is false.

To determine

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 $F=\langle x,0,0\rangle$.

Obtain the divergence.

$\begin{array}{c}\langle \frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\rangle \cdot \langle x,0,0\rangle =\frac{\partial }{\partial x}\left(x\right)+\frac{\partial }{\partial y}\left(0\right)+\frac{\partial }{\partial z}\left(0\right)\\ =1\end{array}$.

Thus, the divergence is 1.

Consider the rotational field $F=\langle 0,y,0\rangle$.

Obtain the divergence.

$\begin{array}{c}\langle \frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\rangle \cdot \langle 0,y,0\rangle =\frac{\partial }{\partial x}\left(0\right)+\frac{\partial }{\partial y}\left(y\right)+\frac{\partial }{\partial z}\left(0\right)\\ =1\end{array}$.

Thus, the divergence is 1.

Hence, the vector fields $F=\langle x,0,0\rangle$ and $F=\langle 0,y,0\rangle$ has same divergence but they don’t differ by a constant vector field.

Thus, the statement is false.

To determine

To explain: Whether the given statement is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Given statement:

If $F=\langle x,y,z\rangle$ and S encloses a region D, then ${\displaystyle \underset{S}{\iint }F\cdot nds}$ is three times the volume of D.

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 ${ℝ}^3$ enclosed by an oriented surface S, then ${\displaystyle \underset{S}{\iint }F\cdot nds}={\displaystyle \underset{D}{\iiint }\nabla \cdot FdV}$, where n is the outward unit normal vector on S.

Calculation:

Consider the rotational field $F=\langle x,y,z\rangle$.

Compute the divergence of F.

$\begin{array}{c}\nabla \cdot F=\frac{\partial }{\partial x}\left(x\right)+\frac{\partial }{\partial y}\left(y\right)+\frac{\partial }{\partial z}\left(z\right)\\ =1+1+1\\ =3\end{array}$

Use the Divergence theorem stated above that obtains the value of ${\displaystyle \underset{S}{\iint }F\cdot nds}$ as follows,

$\begin{array}{c}{\displaystyle \underset{S}{\iint }F\cdot nds}={\displaystyle \underset{D}{\iiint }\nabla \cdot FdV}\\ ={\displaystyle \underset{D}{\iiint }3dV}\\ =3\times \text{volume}\text{of}D\end{array}$

Thus, the statement is true.