Review Exercises for Chapter 7

Question 7.165

Integrate \(f(x, y, z) = {\it xyz}\) along the following paths:

  • (a) \({\bf c}(t)=(e^t\cos t,e^t\sin t, 3),0\leq t\leq 2\pi\)
  • (b) \({\bf c}(t)=(\cos t,\sin t,t),0\leq t\leq 2\pi\)
  • (c) \({\bf c}(t)=\frac{3}{2}t^2{\bf i}+2t^2{\bf j}+t{\bf k},0\leq t\leq 1\)
  • (d) \({\bf c}(t)=t{\bf i}+(1/\sqrt{2})t^2{\bf j}+\frac{1}{3}t^3{\bf k},0\leq t\leq 1\)

Question 7.166

Compute the integral of \(f\) along the path \({\bf c}\) in each of the following cases:

  • (a) \(f(x,y,z)=x+y+yz;{\bf c}(t)=(\sin t,\cos t,t), 0\leq t\leq 2\pi\)
  • (b) \(f(x,y,z)=x+\cos^2 z;{\bf c}(t)=(\sin t,\cos t,t), 0\leq t\leq 2\pi\)
  • (c) \(f(x,y,z)=x+y+z;{\bf c}(t)=(t,t^2,\frac{2}{3}t^3), 0\leq t\leq 1\)

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Question 7.167

Compute each of the following line integrals:

  • (a) \(\int_C(\sin \pi x) \,{\it dy} - (\cos \pi y) \,{\it dz}\), where \(C\) is the triangle whose vertices are \((1, 0, 0), (0, 1, 0)\), and \((0, 0, 1)\), in that order
  • (b) \(\int_C(\sin z) \,{\it dx} +(\cos z) \,{\it dy} -({\it xy})^{1/3} \,{\it dz}\), where \(C\) is the path \({\bf c}(\theta) = (\cos^3\theta\), \(\sin^3\theta, \theta)\), \(0\leq \theta\leq 7\pi/2\)

Question 7.168

If \({\bf F}({\bf x})\) is orthogonal to \({\bf c}'(t)\) at each point on the curve \({\bf x}={\bf c}(t)\), what can you say about \(\int_{\bf c}{\bf F}\, {\cdot} \,d{\bf s}\)?

Question 7.169

Find the work done by the force \({\bf F}(x,y)=(x^2-y^2){\bf i}+2{\it xy}{\bf j}\) in moving a particle counterclockwise around the square with corners \((0, 0), (a, 0), (a, a), (0, a)\), \(a>0\).

Question 7.170

A ring in the shape of the curve \(x^2 + y^2 = a^2\) is formed of thin wire weighing \(|x|+|y|\) grams per unit length at \((x, y)\). Find the mass of the ring.

Question 7.171

Find a parametrization for each of the following surfaces:

  • (a) \(x^2+y^2+z^2-4x-6y=12\)
  • (b) \(2x^2+y^2+z^2-8x=1\)
  • (c) \(4x^2+9y^2-2z^2=8\)

Question 7.172

Find the area of the surface defined by \({\Phi}\colon\, (u,v)\mapsto (x,y,z)\), where \begin{eqnarray*} &&x=h(u,v)=u+v,\qquad y=g(u,v)=u,\\ &&\qquad z=f(u,v)=v;\\[-16pt] \end{eqnarray*} \(0\leq u\leq 1,0\leq v\leq 1\). Sketch.

Question 7.173

Write a formula for the surface area of \({\Phi}\colon\, (r,\theta)\mapsto (x,y,z)\), where \[ x=r\cos\theta,\qquad y=2r\sin\theta,\qquad z=r; \] \(0\leq r\leq 1,0\leq \theta\leq 2\pi\). Describe the surface.

Question 7.174

Suppose \(z = f(x, y)\) and \((\partial f/\partial x)^2+(\partial f/\partial y)^2=c,c>0\). Show that the area of the graph of \(f\) lying over a region \(D\) in the \({\it xy}\) plane is \(\sqrt{1+c}\) times the area of \(D\).

Question 7.175

Compute the integral of \(f(x, y, z) = x^2 + y^2 + z^2\) over the surface in Review Exercise 8.

Question 7.176

Find \({\int\!\!\!\int}_Sf\,dS\) in each of the following cases:

  • (a) \(f(x,y,z)=x;S\) is the part of the plane \(x \,+\, y \,+ \,z = 1\) in the positive octant defined by \(x\geq 0,y\geq 0,z\geq 0\)
  • (b) \(f(x,y,z)=x^2;S\) is the part of the plane \(x = z\) inside the cylinder \(x^2+y^2=1\)
  • (c) \(f(x,y,z)=x;S\) is the part of the cylinder \(x^2 + y^2 = 2x\) with \(0\leq z\leq \sqrt{x^2+y^2}\)

Question 7.177

Compute the integral of \(f(x, y, z) = {\it xyz}\) over the rectangle with vertices \((1, 0, 1), (2, 0, 0)\), \((1, 1, 1)\), and \((2, 1, 0)\).

Question 7.178

Compute the integral of \(x + y\) over the surface of the unit sphere.

Question 7.179

Compute the surface integral of \(x\) over the triangle with vertices \((1, 1, 1), (2, 1, 1)\), and \((2, 0, 3)\).

Question 7.180

A paraboloid of revolution \(S\) is parametrized by \({\Phi}(u,v)=(u\cos v,u\sin v,u^2)\), \(0\leq u\leq 2,0\leq v\leq 2\pi\).

  • (a) Find an equation in \(x, y\), and \(z\) describing the surface.
  • (b) What are the geometric meanings of the parameters \(u\) and \(v\)?
  • (c) Find a unit vector orthogonal to the surface at \({\Phi}(u,v)\).
  • (d) Find the equation for the tangent plane at \({\Phi}(u_0,v_0)=(1,1,2)\) and express your answer in the following two ways:
    • (i) parametrized by \(u\) and \(v \); and
    • (ii) in terms of \(x, y\), and \(z\).
  • (e) Find the area of \(S\).

Question 7.181

Let \(f(x,y,z)=xe^y\cos \pi z\).

  • (a) Compute \({\bf F}= \nabla\! f\).
  • (b) Evaluate \(\int_C{\bf F}\, {\cdot} \,d{\bf s}\), where \({\bf c}(t) = (3 \cos^4 t, 5 \sin^7 t, 0), 0\leq t\leq \pi\).

Question 7.182

Let \({\bf F}(x,y,z)=x{\bf i}+y{\bf j}+z{\bf k}\). Evaluate \({\int\!\!\!\int}_S{\bf F}\, {\cdot}\, \,d{\bf S}\), where \(S\) is the upper hemisphere of the unit sphere \(x^2 + y^2 + z^2=1\).

Question 7.183

Let \({\bf F}(x,y,z)=x{\bf i}+y{\bf j}+z{\bf k}\). Evaluate \(\int_{\bf c} {\bf F}\, {\cdot} \,\,d{\bf s}\), where \({\bf c}(t)=(e^t,t,t^2),0\leq t\leq 1\).

Question 7.184

Let \({\bf F}=\nabla\! f\) for a given scalar function. Let \({\bf c}(t)\) be a closed curve, that is, \({\bf c}(b) = {\bf c}(a)\). Show that \(\int_{\bf c}{\bf F}\, {\cdot} \,d{\bf s} =0\).

Question 7.185

Consider the surface \({\Phi}(u,v)=(u^2\cos v,u^2\sin v,u)\). Compute the unit normal at \(u=1, v = 0\). Compute the equation of the tangent plane at this point.

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Question 7.186

Let \(S\) be the part of the cone \(z^2 = x^2 + y^2\) with \(z \) between 1 and 2 oriented by the normal pointing out of the cone. Compute \({\int\!\!\!\int}_S{\bf F}\, {\cdot}\, d {\bf S}\), where \({\bf F}(x,y,z)=(x^2,y^2,z^2)\).

Question 7.187

Let \({\bf F}=x{\bf i}+x^2{\bf j}+yz{\bf k}\) represent the velocity field of a fluid (velocity measured in meters per second). Compute how many cubic meters of fluid per second are crossing the \({\it xy}\) plane through the square \(0\leq x\leq1,0\leq y\leq 1\).

Question 7.188

Show that the surface area of the part of the sphere \(x^2 + y^2 + z^2 = 1\) lying above the rectangle \([-a,a]\times [-a,a]\), where \(2a^2 < 1\), in the \({\it xy}\) plane is \[ A=2\int^a_{-a}\sin^{-1}\bigg(\frac{a}{\sqrt{1-x^2}}\bigg)\,{\it dx}. \]

Question 7.189

Let \(S\) be a surface and \(C\) a closed curve bounding \(S\). Verify the equality \[ \int\!\!\!\int\nolimits_{\! S} (\nabla \times {\bf F})\, {\cdot}\,d{\bf S} =\int_C{\bf F}\, {\cdot}\,d{\bf s} \] if \({\bf F}\) is a gradient field (use Review Exercise 20).

Question 7.190

Calculate \({\int\!\!\!\int}_S{\bf F}\, {\cdot} \,d{\bf S}\), where \({\bf F}(x,y,z)=(x,y,-y)\) and \(S\) is the cylindrical surface defined by \(x^2+y^2=1, 0\leq z\leq 1\), with normal pointing out of the cylinder.

Question 7.191

Let \(S\) be the portion of the cylinder \(x^2 + y^2 = 4\) between the planes \(z = 0\) and \(z = x + 3\). Compute the following:

  • (a) \({\int\!\!\!\int}_S x^2\,dS\)
  • (b) \({\int\!\!\!\int}_S y^2 \,dS\)
  • (c) \({\int\!\!\!\int}_S z^2 \,dS\)

Question 7.192

Let \(\Gamma\) be the curve of intersection of the plane \(z = ax + by\), with the cylinder \(x^2 + y^2 = 1\). Find all values of the real numbers \(a\) and \(b\) such that \(a^2 + b^2 = 1\) and \[ \int_{\Gamma} y \,{\it dx} +(z-x) \,{\it dy} -y \,{\it dz} =0. \]

Question 7.193

A circular helix that lies on the cylinder \(x^2 + y^2 = R^2\) with pitch \(p\) may be described parametrically by \[ x=R\cos \theta,\qquad y=R\sin \theta,\qquad z=p\theta,\qquad \theta \geq 0. \]

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 \(z,0\leq z <z_0\), 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, and \(t\) is time.

  • (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 \(z=0\).

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1John Nash is the subject of Sylvia Nasar’s best-selling biography, A Beautiful Mind, a fictionalized version of which was made into a movie in 2001.

2We thank Tanya Leise for suggesting this exercise.

3If \({\bf c}\) does not intersect itself [that is, if \({\bf c}(t_1) = {\bf c}(t_2)\) implies \(t_1 = t_2 \)], then each point P on \(C\) (the image curve of \({\bf c} \)) can be written uniquely as \({\bf c}(t)\) for some \(t\). If we define \(f({\rm P}) = f({\bf c}(t)) = {\bf F}({\bf c})\, {\cdot}\, {\bf T}(t)\), \(f\) is a function on \(C\); by definition, its path integral along \({\bf c}\) is given by formula (1) and there is no difficulty in literally interpreting \(\int_{\bf c}{\bf F}\,{\cdot}\, d{\bf s}\) as a path integral. If \({\bf c}\) intersects itself, we cannot define \(f\) as a function on \(C\) as before (why?); however, in this case it is still useful to think of the right side of formula (1) as a path integral.

4See Section 8.5 for a brief discussion of the general theory of differential forms.

5We have not proved that any two one-to-one paths \({\bf c}\) and \({\bf p}\) with the same image must be reparametrizations of each other, but this technical point will be omitted.

6The discovery that electric currents produce magnetic effects was made by Haas Christian Oersted circa 1820. See any elementary physics text for discussions of the physical basis of these ideas.

7Strictly speaking, regularity depends on the parametrization \({\Phi}\) and not just on its image \(S\). Therefore, this terminology is somewhat imprecise; however, it is descriptive and should not cause confusion. (See Exercise 19.)

8In Section 3.5 we showed that level surfaces \(f(x, y, z) = 0\) were in fact graphs of functions of two variables in some neighborhood of a point (\(x_{0}\), \(y_{0}\), \(z_{0})\) satisfying \(\nabla f (x_{0}\), \(y_{0}\), \(z_{0}) \ne 0\). This united two concepts of a surface—graphs and level sets. Again, using the implicit function theorem, it is likewise possible to show that the image of a parametrized surface \({\Phi}\) in the neighborhood of a point (\(u_{0}\), \(v_{0})\) where \({\bf T}_{u} \times {\bf T}_{v} \ne {0}\) is also the graph of a function of two variables. Thus, all definitions of a surface are consistent. (See Exercise 20.)

9Recall from one-variable calculus that \(\cosh u=(e^u + e^{-u})/2\) and \(\sinh u=(e^u - e^{-u})/2\). We easily verify from these definitions that \(\cosh^2 u - \sinh^2 u=1\).

10As we have not yet discussed the independence of parametrization, it may seem that \(A(S)\) depends on the parametrization \({\Phi }\). We shall discuss independence of parametrization in Section 7.7; the use of this notation here should not cause confusion.

11Dirichlet’s functional played a major role in the mathematics of the nineteenth century. The mathematician Georg Friedrich Bernhard Riemann (1826–1866) used it to develop his complex function theory and to give a proof of the famous Riemann mapping theorem. Today it is still used extensively as a tool in the study of partial differential equations.

12We use the term “side” in an intuitive sense. This concept can be developed rigorously, but this will not be done here. Also, the choice of the side to be named the “outside” is often dictated by the surface itself, as, for example, is the case with a sphere. In other cases, the naming is somewhat arbitrary (see the piece of surface depicted in Figure 7.42, for instance).

13There is a single parametrization obtained by cutting a strip of paper, twisting it, and gluing the ends, but it produces a discontinuous \({\bf n}\) on the surface.

14Sometimes one sees the formula \(F=(1/4\pi \varepsilon_0)QQ_0/R^2\). The extra constant \(\varepsilon_0\) appears when MKS units are used for measuring charge. We are using CGS, or Gaussian, units.

15Technically speaking, \(K(p)\) and \(H(p)\) could, in principle, depend on the parametrization \({\Phi}\) of \(S\), but we can show that they are, in fact, independent of \({\Phi}\).

16Space–time is locally like \({\mathbb R}^4\) with three space coordinates and one time coordinate.

17Roughly speaking, this means that \(S\) can be obtained from the sphere by bending and stretching (like with a balloon) but not tearing (the balloon bursts!).

18See C. Misner, K. Thorne, and A. Wheeler, Gravitation, Freeman, New York, 1972.

19Gauss proved that conformal parametrization of a surface always exists. The result of this exercise remains valid even if \({\Phi }\) is not conformal, but the proof is more difficult.

*The solution to this problem may be somewhat time-consuming.

*This section can be skipped on a first reading without loss of continuity.