Question 14.169

\({\bf F}(x,y)=(2,2)\)

Question 14.170

\({\bf F}(x,y)=(4,0)\)

Question 14.171

\({\bf F}(x,y)=(x,y)\)

Question 14.172

\({\bf F}(x,y)=(-x,y)\)

Question 14.173

\({\bf F}(x,y)=(2y,x)\)

Question 14.174

\({\bf F}(x,y)=(y,-2x)\)

Question 14.175

\({\bf F}(x,y)=\bigg(\displaystyle\frac{x}{\sqrt{x^2+y^2}},\displaystyle\frac{y}{\sqrt{x^2+y^2}}\bigg)\)

Question 14.176

\({\bf F}(x,y)=\bigg(\displaystyle\frac{y}{\sqrt{x^2+y^2}},\displaystyle\frac{x}{\sqrt{x^2+y^2}}\bigg)\)

Question 14.177

• (a) \(\textbf{V}(x,y) = x \textbf{i} + y \textbf{j}\)
• (b) \(\textbf{V}(x,y) = y \textbf{i} - x \textbf{j}\)

Question 14.178

• (a) \(\textbf{V}(x,y) = \displaystyle\frac{y}{\sqrt{x^2 + y^2}} \textbf{i} - \displaystyle\frac{x}{\sqrt{x^2 + y^2}} \textbf{j}\)
• (b) \(\textbf{V}(x,y) = \displaystyle\frac{x}{\sqrt{x^2 + y^2}} \textbf{i} + \displaystyle\frac{y}{\sqrt{x^2 + y^2}} \textbf{j}\)

Where are these vector fields not defined? How are these vector fields related to those in Problem 9?

Question 14.179

\({\bf F}(x,y)=(y,-x)\)

Question 14.180

\({\bf F}(x,y)=(x,-y)\)

Question 14.181

\({\bf F}(x,y)=(x,x^2)\)

Question 14.182

\({\bf F}(x,y,z)=(y,-x,0)\)

Question 14.183

\({\bf c}(t)=(e^{2t},\log |t|,1/t),t\neq 0; {\bf F}(x,y,z)=(2x,z,-z^2)\)

Question 14.184

\({\bf c}(t)=(t^2,2t-1,\sqrt{t}),t > 0 ;{\bf F}(x,y,z)=(y+1,2,1/2z)\)

Question 14.185

\({\bf c}(t)=(\sin t,\,\cos t,e^t);{\bf F}(x,y,z)=(y,-x,z)\)

Question 14.186

\({\bf c}(t)=(\displaystyle\frac{1}{t^3},e^t,\displaystyle\frac{1}{t});{\bf F}(x,y,z)=(-3z^4,y,-z^2)\)

Question 14.187

Let \(\textbf{F}(x,y,z)=(x^2,yx^2,z+zx)\) and \(\textbf{c}(t)= (\displaystyle\frac{1}{1-t},0,\displaystyle\frac{e^t}{1-t})\). Show \(\textbf{c}(t)\) is a flow line for \(\textbf{F}\).

Question 14.188

Show that \(\textbf{c}(t)=(a\cos t-b \sin t, a \sin t +b\cos t)\) is a flow line for \(\textbf{F}(x, y)=(-y, x)\) for all real values of \(a\) and \(b\).

Question 14.189

• (a) Let \(\textbf{F}(x, y, z)= (yz, xz, xy)\). Find a function \(f \colon \mathbb R^3 \to \mathbb R\) such that \(\textbf{F}=\nabla f\).
• (b) Let \(\textbf{F}(x, y, z)= (x, y, z)\). Find a function \(f \colon \mathbb R^3 \to \mathbb R\) such that \(\textbf{F}=\nabla f\).

Question 14.190

Let \(f(x, y)=x^2+y^2\). Sketch the gradient vector field \(\nabla f\) together with some level sets of \(f\). How are they related?

Question 14.191

Show that it takes half as much energy to launch a satellite into an orbit just above the earth as it does to escape the earth. (Ignore the rotation of the earth.)

Question 14.192

Let \({\bf c}(t)\) be a flow line of a gradient field \({\bf F}=-{\nabla} V\). Prove that \(V({\bf c}(t))\) is a decreasing function of \(t\).

Question 14.193

Suppose that the isotherms in a region are all concentric spheres centered at the origin. Prove that the energy flux vector field points either toward or away from the origin.

Question 14.194

Sketch the gradient field \(-{ \nabla} V\) for \(V(x,y)=(x+y)/(x^2+y^2)\) and the equipotential surface \(V=1\).

Question 14.195

Let \(\textbf{F}(x, y, z)= (xe^y, y^2z^2, xyz)\) and suppose \(\textbf{c}(t)=\big(x(t), y(t), z(t)\big)\) is a flow line for \(\textbf{F}\). Find the system of differential equations that the functions \(x(t), y(t),\) and \(z(t)\) must satisfy.