Question 4.121

\({\bf c}(t)=(t^3+1,e^{-t},\cos\, (\pi t/2)), \hbox{at } t=1\)

Question 4.122

\({\bf c}(t)=(t^2-1,\cos\, (t^2),t^4), \hbox{at }t=\sqrt{\pi}\)

Question 4.123

\({\bf c}(t)=(e^t,\sin t,\cos t), \hbox{at }t=0\)

Question 4.124

\({\bf c}(t)=\displaystyle\frac{t^2}{1+t^2}{\bf i}+t{\bf j}+ {\bf k}, \hbox{at }t=2\)

Question 4.125

Calculate the tangent and acceleration vectors for the helix \({\bf c}(t)=(\cos t,\sin t ,t)\) at \(t=\pi/4\).

Question 4.126

Calculate the tangent and acceleration vector for the cycloid \({\bf c}(t)=(t-\sin t,1-\cos t)\) at \(t=\pi/4\) and sketch.

Question 4.127

Let a particle of mass \(m\) move on the path \({\bf c}(t) = (t^2,\sin t,\cos t).\) Compute the force acting on the particle at \(t=0\).

Question 4.128

• (a) Let \({\bf c}(t)\) be a path with \(\|{\bf c}(t)\|=\) constant; that is, the curve lies on a sphere. Show that \({\bf c}'(t)\) is orthogonal to \({\bf c}(t)\).
• (b) Let \({\bf c}\) be a path whose speed is never zero. Show that \({\bf c}\) has constant speed if and only if the acceleration vector \({\bf c}''\) is always perpendicular to the velocity vector \({\bf c}'\).

Question 4.129

Let \(\textbf{c}(t)=(\cos t, \sin t, \sqrt3 t)\) be a path in \(\mathbb R^3\).

• (a) Find the velocity and acceleration of this path.
• (b) Find a parametrization for the tangent line to this path at \(t=0\).
• (c) Find the arc length of this path for \(t\in [0, 2\pi]\).

Question 4.130

Let \(\textbf{F}(x, y, z)=(\sin(xz), e^{xy}, x^2y^3z^5)\).

• (a) Find the divergence of \(\textbf{F}\).
• (b) Find the curl of \(\textbf{F}\).

Question 4.131

Verify that the gravitational force field \(\textbf{F}(x, y, z)=-A\displaystyle\frac{(x, y, z)}{(x^2+y^2+z^2)^{3/2}}\), where \(A\) is some constant, is curl free away from the origin.

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Question 4.132

Show that the vector field \(\textbf{V}(x, y, z)=2x\textbf{i} -3y\textbf{j}+4z\textbf{k}\) is not the curl of any vector field.

Question 4.133

Express the arc length of the curve \(x^2 =y^3 =z^5\) between \(x=1\) and \(x=4\) as an integral, using a suitable parametrization.

Question 4.134

Find the arc length of \({\bf c}(t)=t{\bf i}+(\log t){\bf j}+2\sqrt{2t}{\bf k}\) for \(1\leq t\leq 2\).

Question 4.135

A particle is constrained to move around the unit circle in the \(xy\) plane according to the formula \((x,y,z)=(\cos\, (t^2),\sin \,(t^2),0),t\geq 0\).

• (a) What are the velocity vector and speed of the particle as functions of \(t\)?
• (b) At what point on the circle should the particle be released to hit a target at \((2,0,0)\)? (Be careful about which direction the particle is moving around the circle.)
• (c) At what time \(t\) should the release take place? (Use the smallest \(t > 0\) that will work.)
• (d) What are the velocity and speed at the time of release?
• (e) At what time is the target hit?

Question 4.136

A particle of mass \(m\) moves under the influence of a force \({\bf F}=-k{\bf r}\), where \(k\) is a constant and \({\bf r}(t)\) is the position of the particle at time \(t\).

• (a) Write down differential equations for the components of \({\bf r}(t)\).
• (b) Solve the equations in part (a) subject to the initial conditions \({\bf r}(0)={\bf 0}, {\bf r}'(0)=2{\bf j}+{\bf k}\).

Question 4.137

Write the curve described by the equations \(x-1=2y+1=3z+2\) in parametric form.

Question 4.138

Write the curve \(x=y^3=z^2+1\) in parametric form.

Question 4.139

Show that \({\bf c}(t)=(1/(1-t),0,e^t/(1-t))\) is a flow line of the vector field defined by \({\bf F}(x,y,z)=(x^2,0,\) \(z(1+x))\).

Question 4.140

Let \({\bf F}(x,y)=f(x^2+y^2)[-y{\bf i}+x{\bf j}]\) for a function \(f\) of one variable. What equation must \(g(t)\) satisfy for \[ {\bf c}(t)=[\cos g(t)]{\bf i}+[\sin g(t)]{\bf j} \] to be a flow line for \({\bf F}\)?

Question 4.141

\({\bf F}=2x{\bf i}+3y{\bf j}+4z{\bf k}\)

Question 4.142

\({\bf F}=x^2{\bf i}+y^2{\bf j}+z^2{\bf k}\)

Question 4.143

\({\bf F}=(x+y){\bf i}+(y+z){\bf j}+(z+x){\bf k}\)

Question 4.144

\({\bf F}=x{\bf i}+3xy{\bf j}+z{\bf k}\)

Question 4.145

\({\bf F}(x,y,z)=y{\bf i}+z{\bf j}+x{\bf k}\), at the point (1, 1, 1)

Question 4.146

\({\bf F}(x,y,z)=(x+y)^3{\bf i}+(\sin xy){\bf j}+(\cos xyz){\bf k}\), at the point (2, 0, 1)

Question 4.147

\(f(x,y)=e^{xy}+\cos\, (xy)\)

Question 4.148

\(f(x,y)=\displaystyle\frac{x^2-y^2}{x^2+y^2}\)

Question 4.149

\(f(x,y)=e^{x^2}-\cos\, (xy^2)\)

Question 4.150

\(f(x,y)=\tan^{-1}\,(x^2+y^2)\)

Question 4.151

• (a) Let \(f(x,y,z) = xyz^2;\) compute \(\nabla f\).
• (b) Let \({\bf F}(x,y,z) =xy{\bf i}+yz{\bf j}+zy{\bf k}\); compute \(\nabla \times {\bf F}\).
• (c) Compute \(\nabla \times (f {\bf F})\) using identity 10 of the list of vector identities. Compare with a direct computation.

Question 4.152

• (a) Let \({\bf F}=2xye^z{\bf i}+e^zx^2{\bf j}+(x^2ye^z+z^2){\bf k}\). Compute \(\nabla\,{\cdot}\, {\bf F}\) and \(\nabla \times {\bf F}.\)
• (b) Find a function \(f(x,y,z)\) such that \({\bf F}=\nabla f\).

Question 4.153

Let \({\bf F}(x,y)=f(x^2+y^2)[-y{\bf i}+x{\bf j}]\), as in Exercise 20. Calculate div \({\bf F}\) and curl \({\bf F}\) and discuss your answers in view of the results of Exercise 20.

Question 4.154

Let a particle of mass \(m\) move along the elliptical helix \({\bf c}(t)\,{=}\,(4 \cos t, \sin t, t)\).

• (a) Find the equation of the tangent line to the helix at \(t=\pi /4\).
• (b) Find the force acting on the particle at time \(t=\pi /4\).
• (c) Write an expression (in terms of an integral) for the arc length of the curve \({\bf c}({t})\) between \(t= 0\) and \(t=\pi /4\).

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Question 4.155

• (a) Let \(g(x, y, z)=x^{3} + 5{\it yz} + z^{2}\) and let \(h(u)\) be a function of one variable such that \(h' (1) = 1/2\). Let \(f=h \circ g\). Starting at (1, 0, 0), in what directions is \(f\) changing at 50&percent; of its maximum rate?
• (b) For \(g(x, y, z)=x^{3} + 5{\it yz} + z^{2}\), calculate \({\bf F}\,{=}\,\nabla g\), the gradient of \(g\), and verify directly that \(\nabla \times {\bf F} = 0\) at each point (\(x, y, z)\).

Question 4.156

• (a) Write in parametric form the curve that is the intersection of the surfaces \(x^2+y^2+z^2=3\) and \(y=1\).
• (b) Find the equation of the line tangent to this curve at (1, 1, 1).
• (c) Write an integral expression for the arc length of this curve. What is the value of this integral?

Question 4.157

In meteorology, the negative pressure gradient \({\bf G}\) is a vector quantity that points from regions of high pressure to regions of low pressure, normal to the lines of constant pressure (isobars).

• (a) In an \(xy\) coordinate system, \[ {\bf G}=-\frac{\partial P}{\partial x}{\bf i}-\frac{\partial P}{\partial y}{\bf j}. \] Write a formula for the magnitude of the negative pressure gradient.
• (b) If the horizontal pressure gradient provided the only horizontal force acting on the air, the wind would blow directly across the isobars in the direction of \({\bf G}\), and for a given air mass, with acceleration proportional to the magnitude of \({\bf G}\). Explain, using Newton’s second law.
• (c) Because of the rotation of the earth, the wind does not blow in the direction that part (b) would suggest. Instead, it obeys Buys–Ballot’s law, which states: “If in the Northern Hemisphere, you stand with your back to the wind, the high pressure is on your right and the low pressure is on your left.” Draw a figure and introduce \(xy\) coordinates so that \({\bf G}\) points in the proper direction.
• (d) State and graphically illustrate Buys–Ballot’s law for the Southern Hemisphere, in which the orientation of high and low pressure is reversed.

Question 4.158

A sphere of mass \(m\), radius \(a\), and uniform density has potential \(u\) and gravitational force \({\bf F}\), at a distance \(r\) from the center (0, 0, 0), given by \begin{eqnarray*} u=\frac{3m}{2a}-\frac{mr^2}{2a^3},{\bf F}&=&-\frac{m}{a^3}{\bf r}\qquad(r\leq a);\\[6pt] u=\frac{m}{r},{\bf F}&=&-\frac{m}{r^3}{\bf r}\qquad(r >a).\\[-11pt] \end{eqnarray*} Here, \(r=\|{\bf r}\|,{\bf r}=x{\bf i}+y{\bf j}+ z{\bf k}\).

• (a) Verify that \({\bf F}=\nabla u\) on the inside and outside of the sphere.
• (b) Check that \(u\) satisfies Poisson’s equation: \(\partial^2 u/\partial x^2+ \partial^2 u/\partial y^2+\partial^2 u/\partial z^2=\) constant inside the sphere.
• (c) Show that \(u\) satisfies Laplace’s equation: \(\partial^2 u/\partial x^2+\partial^2 u/\partial y^2+\partial^2 u/\partial z^2=0\) outside the sphere.

Question 4.159

A circular helix that lies on the cylinder \(x^2+y^2=R^2\) with pitch \(\rho\) may be described parametrically by \[ x=R\cos \theta,\qquad y=R\sin \theta,\qquad z=\rho \theta,\qquad \theta \geq 0. \] A particle slides under the action of gravity (which acts parallel to the \(z\) axis) without friction along the helix. If the particle starts out at the height \(z_0 >0\), then when it reaches the height \(z\) along the helix, its speed is given by \[ \frac{{\it ds}}{{\it dt}}=\sqrt{(z_0-z)2g}, \] where \(s\) is arc length along the helix, \(g\) is the constant of gravity, \(t\) is time, and \(0 \leq z\leq z_0\).

• (a) Find the length of the part of the helix between the planes \(z=z_0\) and \(z=z_1,0\leq z_1 <z_0\).
• (b) Compute the time \(T_0\) it takes the particle to reach the plane \(z=0\).

Question 4.160

A sphere of radius 10 centimeters (cm) with center at \((0,\,0,\,0)\) rotates about the \(z\) axis with angular velocity 4 in such a direction that the rotation looks counterclockwise from the positive \(z\) axis.

• (a) Find the rotation vector \({\omega}\) (see Example 9, in Section 4.5).
• (b) Find the velocity \({\bf v}={\omega} \times {\bf r}\) when \({\bf r}=5\sqrt{2}({\bf i}-{\bf j})\) is on the “equator.”
• (c) Find the velocity of the point (0, \(5\sqrt{3}\), 5) on the sphere.

Question 4.161

Find the speed of the students in a classroom located at a latitude 49\(^{\circ}\)N due to the rotation of the earth. (Ignore the motion of the earth about the sun, the sun in the galaxy, etc.; the radius of the earth is 3960 miles.)