9.5E: EXERCISES

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Exercise \(\PageIndex{1}\): True or False

For the following exercises, determine whether the statement is true or false.

1. If the coordinate functions of \(\vecs{F}: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) have continuous second partial derivatives, then \(curl \, (div (F))\) equals zero.

2. \(\nabla \cdot (x \, \hat{ \mathbf i} + y\, \hat{ \mathbf j}+ z\, \hat{ \mathbf k}) = 1\).

Answer: False

3. All vector fields of the form \(\vecs{F}(x,y,z) = f(x) \, \hat{ \mathbf i}+ g(y) \, \hat{ \mathbf j} + h(z)\, \hat{ \mathbf k}\) are conservative.

4. If \(curl \, \vecs{F}= 0\), then\( \vecs{F}\) is conservative.

Answer: True

5. If \( \vecs{F}\) is a constant vector field then \(div \,(\vecs{F})= 0\).

6. If \( \vecs{F}\) is a constant vector field then \(curl \,( \vecs{F})= 0\).

Answer: True

Exercise \(\PageIndex{2}\): Curl

For the following exercises, find the curl of \(\vecs{F}.\)

\(\vecs{F}(x,y,z) = xy^2z^4 \, \hat{ \mathbf i} + (2x^2y + z)\, \hat{ \mathbf j} + y^3 z^2 \, \hat{ \mathbf k}\)
\(\vecs{F}(x,y,z) = x^2 z \, \hat{ \mathbf i} + y^2 x\, \hat{ \mathbf j} + (y + 2z) \, \hat{ \mathbf k}\)

Answer

\(curl \,\( \vecs{F}\)= \, \hat{ \mathbf i} + x^2 \, \hat{ \mathbf j} + y^2 \, \hat{ \mathbf k}\)

3. \(\vecs{F}(x,y,z) = 3xyz^2 \, \hat{ \mathbf i} + y^2 \sin z \, \hat{ \mathbf j} + xe^{2z} \, \hat{ \mathbf k}\)

4. \(\vecs{F}(x,y,z) = x^2 yz \, \hat{ \mathbf i} + xy^2 z\, \hat{ \mathbf j} + xyz^2 \, \hat{ \mathbf k}\)

Answer: \(curl \,\( \vecs{F}\)= (xz^2 - xy^2) \, \hat{ \mathbf i}+ (x^2 y - yz^2)\, \hat{ \mathbf j} + (y^2z - x^2z) \, \hat{ \mathbf k}\)

5. \(\vecs{F}(x,y,z) = (x \, \cos y) \, \hat{ \mathbf i}+ xy^2\, \hat{ \mathbf j}\)

6. \(\vecs{F}(x,y,z) = (x - y) \, \hat{ \mathbf i} + (y - z)\, \hat{ \mathbf j} + (z - x) \, \hat{ \mathbf k}\)

Answer: \(curl \,\( \vecs{F}\)= \, \hat{ \mathbf i} + \, \hat{ \mathbf j} + \, \hat{ \mathbf k}\)

7. \(\vecs{F}(x,y,z) = xyz \, \hat{ \mathbf i} + x^2y^2z^2\, \hat{ \mathbf j} + y^2z^3\, \hat{ \mathbf k}\)

8. \(\vecs{F}(x,y,z) = xy \, \hat{ \mathbf i} + yz\, \hat{ \mathbf j} + xz \, \hat{ \mathbf k}\)

Answer: \(curl \,\( \vecs{F}\)= - y \, \hat{ \mathbf i} - z\, \hat{ \mathbf j} - x\, \hat{ \mathbf k}\)

9. \(\vecs{F}(x,y,z) = x^2 \, \hat{ \mathbf i} + y^2 \, \hat{ \mathbf j} + z^2 \, \hat{ \mathbf k}\)

10. \(\vecs{F}(x,y,z) = ax \, \hat{ \mathbf i} + by\, \hat{ \mathbf j} + c \, \hat{ \mathbf k}\) for constants a, b, c

Answer: \(curl \,\( \vecs{F}\)= 0\)

Exercise \(\PageIndex{3}\): Divergence

For the following exercises, find the divergence of \( \vecs{F}.\)

1. \(\vecs{F}(x,y,z) = x^2 z \, \hat{ \mathbf i} + y^2 x \, \hat{ \mathbf j} + (y + 2z) \, \hat{ \mathbf k}\)

2. \(\vecs{F}(x,y,z) = 3xyz^2 \, \hat{ \mathbf i} + y^2 \sin z \, \hat{ \mathbf j} + xe^2 \, \hat{ \mathbf k}\)

Answer: \(div ( \vecs{F})= 3yz^2 + 2y \, \sin z + 2xe^{2z}\)

3. \(\vecs{F}(x,y) = (\sin x) \, \hat{ \mathbf i} + (\cos y) \, \hat{ \mathbf j}\)

4. \(\vecs{F}(x,y,z) = x^2 \, \hat{ \mathbf i} + y^2 \, \hat{ \mathbf j} + z^2 \, \hat{ \mathbf k}\)

Answer: \(div \,\( \vecs{F}\)= 2(x + y + z)\)

5. \(\vecs{F}(x,y,z) = (x - y) \, \hat{ \mathbf i} + (y - z) \, \hat{ \mathbf j} + (z - x) \, \hat{ \mathbf k}\)

6. \(\vecs{F}(x,y) = \dfrac{x}{\sqrt{x^2+y^2}} \, \hat{ \mathbf i} + \dfrac{y}{\sqrt{x^2+y^2}}\, \hat{ \mathbf j}\)

Answer: \(div \,\( \vecs{F}\)= \dfrac{1}{\sqrt{x^2+y^2}}\)

8. \(\vecs{F}(x,y) = x \, \hat{ \mathbf i}- y\, \hat{ \mathbf j}\)

9. \(\vecs{F}(x,y,z) = ax \, \hat{ \mathbf i} + by\, \hat{ \mathbf j} + c\, \hat{ \mathbf k}\) for constants a, b, c

Answer: \(div \,\( \vecs{F}\)= a + b\)

10. \(\vecs{F}(x,y,z) = xyz \, \hat{ \mathbf i}+ x^2y^2z^2\, \hat{ \mathbf j} + y^2z^3\, \hat{ \mathbf k}\)

11. \(\vecs{F}(x,y,z) = xy \, \hat{ \mathbf i} + yz \, \hat{ \mathbf j} + xz \, \hat{ \mathbf k}\)

Answer: \(div \,\( \vecs{F}\)= x + y + z\)

Exercise \(\PageIndex{4}\): Harmonic

For the following exercises, determine whether each of the given scalar functions is harmonic.

1. \(u(x,y,z) = e^{-x} (\cos y - \sin y)\)

2. \(w(x,y,z) = (x^2 + y^2 + z^2)^{-1/2}\)

Answer: Harmonic

Exercise \(\PageIndex{5}\): CURL

1. If \(\vecs{F}(x,y,z) = 2 \, \hat{ \mathbf i} + 2x \, \hat{ \mathbf j} + 3y \, \hat{ \mathbf k}\) and \(\vecs{G}(x,y,z) = x \, \hat{ \mathbf i} - y \, \hat{ \mathbf j} + z \, \hat{ \mathbf k}\), find \(curl \, (F \times G)\).

2. If \(\vecs{F}(x,y,z) = 2 \, \hat{ \mathbf i} + 2x \, \hat{ \mathbf j} + 3y \, \hat{ \mathbf k}\) and \(\vecs{G}(x,y,z) = x \, \hat{ \mathbf i} - y \, \hat{ \mathbf j} + z \, \hat{ \mathbf k}\), find \(div \, (F \times G)\).

Answer: \(div \, (F \times G) = 2z + 3x\)

3. Find \(div \, F\), given that \(F = \nabla f\), where \(f(x,y,z) = xy^3z^2\).

4. Find the divergence of \( \vecs{F}\) for vector field \(\vecs{F}(x,y,z) = (y^2 + z^2) (x + y) i + (z^2 + x^2)(y + z) \, \hat{ \mathbf j} + (x^2 + y^2)(z + x) \, \hat{ \mathbf k}\).

Answer: \(div \,\( \vecs{F}\)= 2r^2\)

5. Find the divergence of \( \vecs{F}\) for vector field \(\vecs{F}(x,y,z) = f_1(y,z) \, \hat{ \mathbf i} + f_2 (x,z) \, \hat{ \mathbf j} + f_3 (x,y) \, \hat{ \mathbf k}\).

6. For the following exercises, use \(r = |r|\) and \(r = (x,y,z)\).

a) Find the \(curl \, r\)

Answer: \(curl \, r = 0\)

b) Find the \(curl \, \dfrac{r}{r}\).

c) Find the \(curl \, \dfrac{r}{r^3}\).

Answer: \(curl \, \dfrac{r}{r^3} = 0\)

7. Let \(\vecs{F}(x,y) = \dfrac{-y\, \hat{ \mathbf i}+x\, \hat{ \mathbf j}}{x^2+y^2}\), where\( \vecs{F}\)is defined on \(\{(x,y) \in \mathbb{R} | (x,y) \neq (0,0) \}\). Find \(curl \, F\).

Exercise \(\PageIndex{6}\): Curl

For the following exercises, use a computer algebra system to find the curl of the given vector fields.

1. [T] \(\vecs{F}x,y,z) = arctan \left(\dfrac{x}{y}\right) \, \hat{ \mathbf i} + \ln \sqrt{x^2 + y^2} \, \hat{ \mathbf j} + \, \hat{ \mathbf k}\)

Answer: \(curl \,\( \vecs{F}\)= \dfrac{2x}{x^2+y^2} \, \hat{ \mathbf k}\)

2. [T] \(\vecs{F}(x,y,z) = \sin (x - y) \, \hat{ \mathbf i}+ \sin (y - z) \, \hat{ \mathbf j} + \sin (z - x) \, \hat{ \mathbf k}\)

Exercise \(\PageIndex{7}\): Divergence

For the following exercises, find the divergence of\( \vecs{F}\)at the given point.

1. \(\vecs{F}(x,y,z) = \, \hat{ \mathbf i} + \, \hat{ \mathbf j} + \, \hat{ \mathbf k}\) at \((2, -1, 3)\)

Answer: \(div \,( \vecs{F})= 0\)

2. \(\vecs{F}(x,y,z) = xyz \, \hat{ \mathbf i}+ y \, \hat{ \mathbf j} + z\, \hat{ \mathbf k}\) at \((1, 2, 3)\)

3. \(\vecs{F}(x,y,z) = e^{-xy}\, \hat{ \mathbf i} + e^{xz}\, \hat{ \mathbf j} + e^{yz}\, \hat{ \mathbf k}\) at \((3, 2, 0)\)

Answer: \(div \, \vecs{F}= 2 - 2e^{-6}\)

4. \(\vecs{F}(x,y,z) = xyz \, \hat{ \mathbf i} + y \, \hat{ \mathbf j} + z\, \hat{ \mathbf k}\) at \((1, 2, 1)\)

5. \(\vecs{F}(x,y,z) = e^x \sin y \, \hat{ \mathbf i} - e^x \cos y \, \hat{ \mathbf j} \) at \((0, 0, 3)\)

Answer: \(div \,\( \vecs{F}\)= 0\)

Exercise \(\PageIndex{8}\): CURL

For the following exercises, find the curl of\( \vecs{F}\)at the given point.

1. \(\vecs{F}(x,y,z) = \, \hat{ \mathbf i}+ \, \hat{ \mathbf j} + \, \hat{ \mathbf k}\) at \((2, -1, 3)\)

2. \(\vecs{F}(x,y,z) = xyz \, \hat{ \mathbf i} + y \, \hat{ \mathbf j} + z\, \hat{ \mathbf k}\) at \((1, 2, 3)\)

Answer: \(curl \,\( \vecs{F}\)= \, \hat{ \mathbf j} - 3\, \hat{ \mathbf k}\)

3. \(\vecs{F}(x,y,z) = e^{-xy}\, \hat{ \mathbf i} + e^{xz}\, \hat{ \mathbf j} + e^{yz}\, \hat{ \mathbf k}\) at \((3, 2, 0)\)

4. \(\vecs{F}(x,y,z) = xyz\, \hat{ \mathbf i}+ y \, \hat{ \mathbf j} + z\, \hat{ \mathbf k}\) at \((1, 2, 1)\)

Answer: \(curl \,\( \vecs{F}\)= 2\, \hat{ \mathbf j} - \, \hat{ \mathbf k}\

5. \(\vecs{F}(x,y,z) = e^x \sin y \, \hat{ \mathbf i} - e^x \cos y \, \hat{ \mathbf j} \) at \((0, 0, 3)\)

Exercise \(\PageIndex{9}\)

Let \(\vecs{F}(x,y,z) = (3x^2 y + az) \, \hat{ \mathbf i} + x^3 \, \hat{ \mathbf j} + (3x + 3z^2) \, \hat{ \mathbf k}\). For what value of a is\( \vecs{F}\)conservative?

Answer: \(a = 3\)

Exercise \(\PageIndex{10}\)

1. Given vector field \(\vecs{F}(x,y) = \dfrac{1}{x^2+y^2} (-y,x)\) on domain \(D = \dfrac{\mathbb{R}^2}{\{(0,0)\}} = \{(x,y) \in \mathbb{R}^2 |(x,y) \neq (0,0) \}\), is\( \vecs{F}\)conservative?

2. Given vector field \(\vecs{F}(x,y) = \dfrac{1}{x^2+y^2} (x,y)\) on domain \(D = \dfrac{\mathbb{R}^2}{\{(0,0)\}}\), is\( \vecs{F}\)conservative?

Answer: \(\vecs{F}\) is conservative.

3. Find the work done by force field \(\vecs{F}(x,y) = e^{-y}\, \hat{ \mathbf i} - xe^{-y}\, \hat{ \mathbf j}\) in moving an object from P(0, 1) to Q(2, 0). Is the force field conservative?

Exercise \(\PageIndex{11}\)

1. Compute divergence \(\vecs{F}= (\sinh x) \, \hat{ \mathbf i} + (\cosh y) \, \hat{ \mathbf j} - xyz \, \hat{ \mathbf k}\).

Answer: \(div \,( \vecs{F})= \cosh x + \sinh y - xy\)

2. Compute \(curl \,( \vecs{F})= (\sinh x) \, \hat{ \mathbf i}+ (\cosh y) \, \hat{ \mathbf j} - xyz \, \hat{ \mathbf k}\).

Answer: TBA

Exercise \(\PageIndex{12}\)

For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity \(\omega = \langle a,b,c \rangle\). If P is a point in the body located at \(\vecs{r}= x\, \hat{ \mathbf i}+ y\, \hat{ \mathbf j} + z\, \hat{ \mathbf k}\), the velocity at P is given by vector field \(\vecs{F} = \omega \times \vecs{r} \).

A three dimensional diagram of an object rotating about the x axis in a counterclockwise manner with constant angular velocity w = <a,b,c>. The object is roughly a sphere with pointed ends on the x axis, which cuts it in half. An arrow r is drawn from (0,0,0) to P(x,y,z) and down from P(x,y,z) to the x axis.

a) Express\( \vecs{F}\) in terms of \(\, \hat{ \mathbf i}, \, \hat{ \mathbf j},\) and \(\, \hat{ \mathbf k}\) vectors.

Answer: \((bz - cy)\, \hat{ \mathbf i}+(cx - az)\, \hat{ \mathbf j} + (ay - bx)\, \hat{ \mathbf k}\)

b) Find \(div \, \vecs{F}\).

c) Find \(curl \, \vecs{F}\)

Answer: \(curl \,( \vecs{F})= 2\omega\)

Exercise \(\PageIndex{13}\)

In the following exercises, suppose that \(\nabla \cdot \vecs{F}= 0\) and \(\nabla \cdot \vecs{G}= 0\).

a) Does \(\vecs{F} + \vecs{G}\) necessarily have zero divergence?

b) Does \(\vecs{F} \times \vecs{G}\) necessarily have zero divergence?

Answer: \(\vecs{F} \times \vecs{G}\) does not have zero divergence.

Exercise \(\PageIndex{14}\)

In the following exercises, suppose a solid object in \(\mathbb{R}^3\) has a temperature distribution given by \(T(x,y,z)\). The heat flow vector field in the object is \(\vecs{F}= - k \nabla T\), where \(k > 0\) is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \vecs{F}= -k \nabla \cdot \nabla T = - k \nabla^2 T\).

a) Compute the heat flow vector field.

b) Compute the divergence.

Answer: \(\nabla \cdot \vecs{F}= -200 k [1 + 2(x^2 + y^2 + z^2)] e^{-x^2+y^2+z^2}\)

Exercise \(\PageIndex{15}\)

[T] Consider rotational velocity field \(\vecs{v}= \langle 0,10z, -10y \rangle\). If a paddlewheel is placed in plane \(x + y + z = 1\) with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.

A three dimensional diagram of a rotational velocity field. The arrows are showing a rotation in a clockwise manner. A paddlewheel is shown in plan x + y + z = 1 with n extended out perpendicular to the plane.

Glossary

curl

the curl of vector field \(F = \langle P,Q,R \rangle\), denoted \(\nabla \times F\), is the “determinant” of the matrix \(\begin{vmatrix} i \, \hat{ \mathbf j} \, \hat{ \mathbf k} \nonumber \\ \dfrac{\partial}{\partial x} \dfrac{\partial}{\partial y} \dfrac{\partial}{\partial z} \nonumber \\ P Q R \end{vmatrix}\) and is given by the expression \((R_y - Q_z)\, \hat{ \mathbf i} + (P_z - R_x ) \, \hat{ \mathbf j} + (Q_x - P_y) \, \hat{ \mathbf k}\); it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point

divergence

the divergence of a vector field \(F = \langle P,Q,R\rangle \), denoted \(\nabla \times F\), is \(P_x + Q_y + R_z\); it measures the “outflowing-ness” of a vector field