Question 13.250

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

Question 13.251

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

Question 13.252

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

Question 13.253

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

Question 13.254

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

Question 13.255

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

Question 13.256

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

Question 13.257

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

Question 13.258

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

Question 13.259

• (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 13.260

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

Question 13.261

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

Question 13.262

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

Question 13.263

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

Question 13.264

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

Question 13.265

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

Question 13.266

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

Question 13.267

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

Question 13.268

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 13.269

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 13.270

• (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 13.271

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 13.272

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 13.273

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 13.274

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 13.275

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

Question 13.276

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.