Question 8.75

Determine which of the following vector fields \({\bf F}\) in the plane is the gradient of a scalar function \(f\). If such an \(f\) exists, find it.

• (a) \({\bf F} (x,y) = x {\bf i} + y {\bf j}\)
• (b) \({\bf F} (x,y) = xy {\bf i} + xy {\bf j}\)
• (c) \({\bf F} (x,y) = (x^2 + y^2) {\bf i} + 2xy {\bf j}\)

Question 8.76

Repeat Exercise 1 for the following vector fields:

• (a) \({\bf F} (x,y) = ( \cos xy - xy \sin xy) {\bf i} - (x^2 \sin xy) {\bf j}\)
• (b) \({\bf F} (x,y) = (x \sqrt{x^2 y^2 +1}) {\bf i} + (y \sqrt{x^2 y^2+1} ) {\bf j}\)
• (c) \({\bf F} (x,y) = (2x \cos y + \cos y) {\bf i} - (x^2 \sin y + x \sin y) {\bf j}\)

460

Question 8.77

For each of the following vector fields \(\textbf{F}\), determine (i) if there exists a function \(g\) such that \(\nabla g = \textbf{F}\), and (ii) if there exists a vector field \(\textbf{G}\) such that \(\hbox{curl } \textbf{G} = \textbf{F}\). (It is not necessary to find \(g\) or \(\textbf{G}\).)

• (a) \(\textbf{F}(x,y,z) = ( 4xz - x , -4yz , z - 2y )\)
• (b) \(\textbf{F}(x,y,z) = ( e^x \sin y , e^x \cos y , z^2 )\)
• (c) \(\textbf{F}(x,y,z) = ( \log (z^2 + 1) + y^2 , 2xy , \frac{2xz}{z^2 + 1} )\)
• (d) \(\textbf{F}(x,y,z) = ( x^2 + x \sin z\), \(y \cos z - 2xy, \cos z + \sin z )\)

Question 8.78

For each of the following vector fields \(\textbf{F}\), determine (i) if there exists a function \(g\) such that \(\nabla g = \textbf{F}\), and (ii) if there exists a vector field \(\textbf{G}\) such that \(\hbox{curl } \textbf{G} = \textbf{F}\). (It is not necessary to find \(g\) or \(\textbf{G}\).)

• (a) \(\textbf{F}(x,y,z) = ( e^x \cos y , - e^x \sin y , \pi )\)
• (b) \(\textbf{F}(x,y,z) = \left( \frac{y}{z^2 + 4}, \frac{x}{z^2 + 4}, \frac{-2xyz}{z^4 + 8z^2 + 16} \right)\)
• (c) \(\textbf{F}(x,y,z) = ( x^2 y^2 z^2, y e^x , xy \cos z )\)
• (d) \(\textbf{F}(x,y,z) = ( 6z^5 y^5 , 9x^8 z^2 , 4x^3 y^3 )\)

Question 8.79

Show that any two potential functions for a vector field on \({\mathbb R}^3\) differ at most by a constant.

Question 8.80

• (a) Let \({\bf F} (x,y) = (xy, y^2)\) and let \({\bf c}\) be the path \(y = 2x^2\) joining \((0, 0)\) to \((1, 2)\) in \({\mathbb R}^2\). Evaluate \(\int_{\bf c} {\bf F} \,{\cdot}\, d {\bf s}\).
• (b) Does the integral in part (a) depend on the path joining \((0,0)\) to \((1, 2)\)?

Question 8.81

Let \({\bf F} (x,y,z) = (2xyz + \sin x) {\bf i} + x^2 z {\bf j} + x^2 y {\bf k}\). Find a function \(f\) such that \({\bf F} = {\nabla}\! f\).

Question 8.82

Evaluate \(\int_{\bf c} {\bf F} \,{\cdot}\, d {\bf s}\), where \({\bf c} (t) = (\cos^5 t, \sin^3 t, t^4), 0 \le t \le \pi\), and \({\bf F}\) is as in Exercise 7.

Question 8.83

If \(f(x)\) is a smooth function of one variable, must \({\bf F}(x,y)=f(x){\bf i}+f(y){\bf j}\) be a gradient?

Question 8.84

• (a) Show that \({\bf F}=-{\bf r}/\|{\bf r}\|^3\) is the gradient of \(f(x,y,z)=1/r\).
• (b) What is the work done by the force \({\bf F} = - {\bf r} / \| {\bf r}\|^3\) in moving a particle from a point \({\bf r}_0 \in {\mathbb R}^3\) “to \(\infty\),” where \({\bf r} (x,y,z)= (x,y,z)?\)

Question 8.85

Let \({\bf F} (x,y,z) = xy {\bf i}+ y {\bf j} + z {\bf k}\). Can there exist a function \(f\) such that \({\bf F} = {\nabla}\! f\)?

Question 8.86

Let \({\bf F} = F_1 {\bf i} + F_2 {\bf j} + F_3 {\bf k}\) and suppose each \(F_k\) satisfies the homogeneity condition \[ F_k (tx, ty,tz) = t F_k (x,y,z), \qquad k=1,2,3. \]

Suppose also \({\nabla} \times {\bf F} ={\bf 0}\). Prove that \({\bf F} = {\nabla}\! f\), where \[ 2f (x,y,z) = x F_1 (x,y,z) + y F_2 (x,y,z) +z F_3 (x,y,z). \] [HINT: Use Review Exercise 31, Chapter 2.]

Question 8.87

Let \({\bf F} (x,y,z) = (e^x \sin y) {\bf i} + ( e^x \cos y) {\bf j} + z^2 {\bf k}\). Evaluate the integral \(\int_{\bf c} {\bf F} \,{\cdot}\, d {\bf s}\), where \({\bf c} (t) = ( \sqrt{t}, t^3, \exp \sqrt{t}), 0 \le t \le 1\).

Question 8.88

Let a fluid have the velocity field \({\bf F} (x,y,z) = xy {\bf i} + yz {\bf j} + xz {\bf k}\). What is the circulation around the unit circle in the \(xy\) plane? Interpret your answer.

Question 8.89

The mass of the earth is approximately \(6 \times 10^{27}\) g and that of the sun is 330,000 times as much. The gravitational constant is \(6.7 \times 10^{-8} {\rm cm}^3/{\rm s}^2\,{\cdot}\,\)g. The distance of the earth from the sun is about \(1.5 \times 10^{12}\) cm. Compute, approximately, the work necessary to increase the distance of the earth from the sun by 1 cm.

Question 8.90

• (a) Show that \(\int_C ( x\, {\it dy} - y {\it {\,d} x})/ ( x^2 + y^2) = 2 \pi\), where \(C\) is the unit circle.
• (b) Conclude that the associated vector field \([-y / (x^2 + y^2) ] {\bf i} + [x/(x^2 + y^2)] {\bf j}\) is not a conservative field.
• (c) Show, however, that \(\partial\! P / \partial y = \partial\! Q / \partial x\). Does this contradict the corollary to Theorem 7? If not, why not?

Question 8.91

Determine if the following vector fields \(\textbf{F}\) are gradient fields. If there exists a fuction \(f\) such that \(\nabla f = \textbf{F}\), find \(f\).

• (a) \(\textbf{F}(x,y,z) = ( 2xyz , x^2 z , x^2 y )\)
• (b) \(\textbf{F}(x,y) = ( x \cos y , x \sin y )\)
• (c) \(\textbf{F}(x,y,z) = ( x^2 e^y , xyz , e^z )\)
• (d) \(\textbf{F}(x,y) = ( 2x \cos y , - x^2 \sin y )\)

Question 8.92

Determine if the following vector fields \(\textbf{F}\) are gradient fields. If there exists a function \(f\) such that \(\nabla f = \textbf{F}\), find \(f\).

• (a) \(\textbf{F}(x,y) = ( 2x + y^2 - y \sin x , 2xy z + \cos x )\)
• (b) \(\textbf{F}(x,y,z) = ( 6x^2 z^2 , 5x^2 y^2 , 4y^2 z^2 )\)
• (c) \(\textbf{F}(x,y) = ( y^3 + 1 , 3xy^2 + 1 )\)
• (d) \(\textbf{F}(x,y) = ( x e^{(x^2 + y^2)} + 2xy , y e^{(x^2 + y^2)} + 4y^3 z , y^4 )\)

Question 8.93

Show that the following vector fields are conservative. Calculate \(\int_C {\bf F} \,{\cdot}\, d {\bf s}\) for the given curve.

• (a) \({\bf F} = (xy^2 + 3x^2 y) {\bf i}+ ( x+ y) x^2 {\bf j}; C\) is the curve consisting of line segments from \((1,1)\) to \((0,2)\) to \((3,0)\).
• (b) \({\bf F} = {\displaystyle \frac{2x}{y^2 + 1} }{\bf i} - {\displaystyle\frac{2y (x^2 +1)}{(y^2 +1)^2}} {\bf j}\); \(C\) is parametrized by \(x = t^3 -1, y = t^6 -t, 0 \le t \le 1\).
• (c) \({\bf F} = [\cos\, (xy^2) - xy^2 \sin\, (xy^2)] {\bf i} - 2x^2 y \sin\, (xy^2) {\bf j}\); \(C\) is the curve \((e^t, e^{t+1}), -1 \le t \le 0\).

461

Question 8.94

Prove Theorem 8. [HINT: Define \({\bf G} = G_1 {\bf i} + G_2 {\bf j} + G_3 {\bf k}\) by \begin{eqnarray*} G_1 (x,y,z) & = & \int_0^z F_2 (x,y,t) \,{\it {\,d} t} - \int_0^y F_3 (x,t,0) \,{\it {\,d} t} \\[2pt] G_2 (x,y,z) & = & - \int_0^z F_1 (x,y,t) \,{\it {\,d} t}\\[-10pt] \end{eqnarray*} and \(G_3 (x,y,z)=0\).]

Question 8.95

Is each of the following vector fields the curl of some other vector field? If so, find the vector field.

• (a) \({\bf F} = x {\bf i} + y {\bf j} +z {\bf k}\)
• (b) \({\bf F} = (x^2 +1) {\bf i} + (z- 2xy) {\bf j} + y {\bf k}\)

Question 8.96

Let \({\bf F} = xz {\bf i} - yz {\bf j} + y {\bf k}\). Verify that \({\nabla} \,{\cdot}\, {\bf F} =0\). Find a \({\bf G}\) such that \({\bf F} = {\nabla} \times {\bf G}\).

Question 8.97

Repeat Exercise 22 for \({\bf F} = y^2 {\bf i} + z^2 {\bf j} + x^2 {\bf k}\).

Question 8.98

Let \({\bf F} = xe^y {\bf i} - (x \cos z) {\bf j} - z e^y {\bf k}\). Find a \({\bf G}\) such that \({\bf F} = {\nabla} \times {\bf G}\).

Question 8.99

Let \({\bf F} = (x \cos y) {\bf i} - (\sin y) {\bf j} + (\sin x) {\bf k}\). Find a \({\bf G}\) such that \({\bf F} = {\nabla} \times {\bf G}\).

Question 8.100

By using different paths from \((0, 0, 0)\) to \((x,y,z)\), show that the function \(f\) defined in the proof of Theorem 7 for “condition (ii) implies condition (iii)” satisfies \(\partial\! f / \partial x = F_1\) and \(\partial\! f/ \partial y = F_2\).

Question 8.101

Let \({\bf F}\) be the vector field on \({\mathbb R}^3\) given by \({\bf F} = - y {\bf i} + x {\bf j}\).

• (a) Show that \({\bf F}\) is rotational, that is, \({\bf F}\) is not irrotational.
• (b) Suppose \({\bf F}\) represents the velocity vector field of a fluid. Show that if we place a cork in this fluid, it will revolve in a plane parallel to the \(xy\) plane, in a circular trajectory about the \(z\) axis.
• (c) In what direction does the cork revolve?

Question 8.102

Let \({\bf G}\) be the vector field on \({\mathbb R}^3\backslash \{z \hbox{ axis}\}\) defined by \[ {\bf G} = \frac{-y}{x^2 + y^2}\, {\bf i} + \frac{x}{x^2 + y^2}\, {\bf j}. \]

• (a) Show that \({\bf G}\) is irrotational.
• (b) Show that the result of Exercise 27(b) holds for \({\bf G}\) also.
• (c) How can we resolve the fact that the trajectories of \({\bf F}\) and \({\bf G}\) are both the same (circular about the \(z\) axis) yet \({\bf F}\) is rotational and \({\bf G}\) is not? [HINT: The property of being rotational is a local condition, that is, a property of the fluid in the neighborhood of a point.]

Question 8.103

Let \({\bf F} = - ({\it GmM} {\bf r} / r^3)\) be the gravitational force field defined on \({\mathbb R}^3 \backslash \{{\bf 0}\}\).

• (a) Show that div \({\bf F} =0\).
• (b) Show that \({\bf F} \ne \hbox{curl } {\bf G}\) for any \(C^1\) vector field \({\bf G}\) on \({\mathbb R}^3 \backslash \{{\bf 0}\}\).