Integrate \(f(x, y, z) = {\it xyz}\) along the following paths:
Compute the integral of \(f\) along the path \({\bf c}\) in each of the following cases:
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Compute each of the following line integrals:
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 \(\int_{\bf c}{\bf F}\, {\cdot} \,d{\bf s}\)?
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 \((0, 0), (a, 0), (a, a), (0, a)\), \(a>0\).
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.
Find a parametrization for each of the following surfaces:
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.
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.
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 \(D\).
Compute the integral of \(f(x, y, z) = x^2 + y^2 + z^2\) over the surface in Review Exercise 8 (three exercises back).
Find \({\int\!\!\!\int}_Sf\,dS\) in each of the following cases:
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 \((2, 1, 0)\).
Compute the integral of \(x + y\) over the surface of the unit sphere.
Compute the surface integral of \(x\) over the triangle with vertices \((1, 1, 1), (2, 1, 1)\), and \((2, 0, 3)\).
A paraboloid of revolution \(S\) is parametrized by \({\Phi}(u,v)=(u\cos v,u\sin v,u^2)\), \(0\leq u\leq 2,0\leq v\leq 2\pi\).
Let \(f(x,y,z)=xe^y\cos \pi z\).
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 \(x^2 + y^2 + z^2=1\).
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 \({\bf c}(t)=(e^t,t,t^2),0\leq t\leq 1\).
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 \(\int_{\bf c}{\bf F}\, {\cdot} \,d{\bf s} =0\).
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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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 \({\bf F}(x,y,z)=(x^2,y^2,z^2)\).
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 \(0\leq x\leq1,0\leq y\leq 1\).
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 \[ A=2\int^a_{-a}\sin^{-1}\bigg(\frac{a}{\sqrt{1-x^2}}\bigg)\,{\it dx}. \]
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, five exercises back).
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.
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:
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 \[ \int_{\Gamma} y \,{\it dx} +(z-x) \,{\it dy} -y \,{\it dz} =0. \]
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.
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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 18.5 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 14.5 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 17.6; 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 17.42, 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.