Question 8.134

Evaluate \(\omega \wedge \eta\) if

• (a) \(\begin{array}{r@{\,}c@{\,}l}\\ \omega &=& 2x{\it {\,d} x} + y \,{\it dy}\\ \eta &=& x^3 {\it {\,d} x} + y^2 {\it dy}\end{array}\)
• (b) \(\begin{array}{r@{\,}c@{\,}l}\\ \omega &=& x {\it {\,d} x} - y \,{\it dy}\\ \eta &=& y {\it {\,d} x} + x \,{\it dy}\end{array}\)
• (c) \(\begin{array}{r@{\,}c@{\,}l}\\ \omega &=& x {\it {\,d} x} + y\, {\it dy} + z {\,d} z\\ \eta &=& z {\it {\,d} x}\, {\it dy} + x\, {\it dy}\, {\,d} z + y {\,d} z\, {\it {\,d} x}\end{array}\)
• (d) \(\begin{array}{r@{\,}c@{\,}l}\\ \omega &=& x y\, {\it dy}\, {\,d} z + x^2 {\it {\,d} x}\, {\it dy}\\ \eta &=& {\it {\,d} x} + {\,d} z\end{array}\)
• (e) \(\begin{array}{r@{\,}c@{\,}l}\\ \omega &=& e^{xyz} {\it {\,d} x}\, {\it dy} \\ \eta &=& e^{-xyz} {\,d} z\end{array}\)

Question 8.135

Prove that \begin{eqnarray*} &&(a_1{\it {\,d} x}\, + a_2\, {\it dy} + a_3 {\,d} z) \wedge (b_1\,{\it dy}\, {\,d} z + b_2 {\,d} z\,{\it {\,d} x} + b_3 {\it {\,d} x}\,{\it dy})\\[3pt] &&\quad = \left(\sum_{i=1}^3 a_ib_i\right)\!\!{\it {\,d} x}\,{\it dy}\,{\,d} z. \end{eqnarray*}

Question 8.136

Find \({\,d}\omega\) in the following examples:

• (a) \(\omega = x^2 y + y^3\)
• (b) \(\omega = y^2 \cos x\, {\it dy} + xy\, {\it {\,d} x} + {\,d} z\)
• (c) \(\omega = x y\, {\it dy} + (x + y )^2 {\it {\,d} x}\)
• (d) \(\omega = x {\it {\,d} x}\, {\it dy} + z\, {\it dy}\, {\,d} z + y {\,d} z\, {\it {\,d} x}\)
• (e) \(\omega = (x^2 + y^2)\, {\it dy}\, {\,d} z\)
• (f) \(\omega = (x^2 + y^2 + z^2){\,d} z\)
• (g) \(\omega = \displaystyle \frac{-x}{x^2 + y^2}\,{\it {\,d} x} +\frac{y}{x^2 + y^2} {\it dy}\)
• (h) \(\omega = x^2 y\, {\it dy}\, {\,d} z\)

Question 8.137

Let \(C\) be the line segment from the point \((-2,0,1)\) to \((3,6,9)\). Let \(\omega_1 = y\, {\it dx} + x\, {\it dy} + xy\, dz\), \(\omega_2 = z\, {\it dx} + y\, {\it dy} + 2x\, dz\), and \(f(x,y,z) = xy\). Calculate the following:

• (a) \(\int_{C} f \omega_1\)
• (b) \(\int_{C} f \omega_2\)
• (c) \(\int_{C} \omega_1 + \omega_2\)

Question 8.138

Let \(C\) be parameterized by \(c(t) = (t^2 + 4t , \, t + 1), t \in [0, \pi]\). Let \(\omega_1 = y\, {\it dx} + x\, {\it dy}\), \(\omega_2 = y^2 {\it dx} + x^2 {\it dy}\), and \(f(x,y) = x\). Calculate the following: \[ {\rm (a)} \int_{C} f \omega_1 {\rm (b)} \int_{C} f \omega_2 {\rm (c)} \int_{C} \omega_1 + \omega_2 \]

Question 8.139

Let \({\bf V}\colon\, K\to {\mathbb R}^3\) be a vector field defined by \({\bf V}(x,y,z) = G(x,y,z){\bf i} + H(x,y,z){\bf j} + F(x,y,z){\bf k}\), and let \(\eta\) be the 2-form on \(K\) given by \[ \eta = F {\it {\,d} x}\, {\it dy} + G\, {\it dy}\, {\,d} z + H {\,d} z\, {\it {\,d} x}. \]

Show that \({\,d} \eta = (\hbox{div }{\bf V})\,{\it {\,d} x}\, {\it dy}\, {\,d} z\).

Question 8.140

If \({\bf V} = A(x,y,z){\bf i} + B(x,y,z){\bf j} + C(x,y,z){\bf k}\) is a vector field on \(K \subset {\mathbb R}^3\), define the operation \({\rm Form}_2\): \(\hbox{Vector Fields} \to \hbox{2-forms}\) by \[ \hbox{Form}_2({\bf V}) = A\, {\it dy}\, {\,d} z + B {\,d} z\, {\it {\,d} x} + C {\it {\,d} x}\, {\it dy}. \]

• (a) Show that \({\rm Form}_2(\alpha{\bf V}_1 + {\bf V}_2)=\alpha\hbox{ Form}_2\, ({\bf V}_1) + \hbox{Form}_2({\bf V}_2)\), where \(\alpha\) is a real number.
• (b) Show that \({\rm Form}_2 (\hbox{curl }{\bf V}) = {\,d} \omega\), where \(\omega =A{\it {\,d} x} + B\,{\it dy} + C{\,d} z\).

Question 8.141

Using the differential form version of Stokes’ theorem, prove the vector field version in Section 8.3. Repeat for Gauss’ theorem.

Question 8.142

Interpret Theorem 16 in the case \(k = 1\).

Question 8.143

Let \(\omega = (x+ y){\,d} z + (y + z)\,{\it {\,d} x} + (x+ z)\, {\it dy}\), and let \(S\) be the upper part of the unit sphere; that is, \(S\) is the set of \((x,y,z)\) with \(x^2 + y^2 + z^2 = 1\) and \(z \ge 0\). \(\partial\! S\) is the unit circle in the \(xy\) plane. Evaluate \(\int_{\partial S}\omega\) both directly and by Stokes’ theorem.

Question 8.144

Let \(T\) be the triangular solid bounded by the \(xy\) plane, the \(xz\) plane, the \(yz\) plane, and the plane \(2x + 3y + 6z = 12\). Compute \[ \intop\!\!\!\intop\nolimits_{{\partial\! T}} F_1\, {\it {\,d} x}\, {\it dy} + F_2\, {\it dy}\, {\,d} z + F_3 {\,d} z\, {\it {\,d} x} \] directly and by Gauss’ theorem, if

• (a) \(F_1 = 3 y , \ F_2 = 18z, \ F_3 = -12\); and
• (b) \(F_1 = z,\ F_2 = x^2,\ F_3 = y\).

Question 8.145

Evaluate \({\intop\!\!\!\intop}_S\omega\), where \(\omega = z {\it {\,d} x}\, {\it dy} + x\, {\it dy}\, {\,d} z + y {\,d} z\, {\it {\,d} x}\) and \(S\) is the unit sphere, directly and by Gauss’ theorem.

Question 8.146

Let \(R\) be an elementary region in \({\mathbb R}^3\). Show that the volume of \(R\) is given by the formula \[ v(R) = \frac13 \intop\!\!\!\intop\nolimits_{{\partial\! R}} x\, {\it dy}\, {\,d} z + y {\,d} z\, {\it {\,d} x} + z\, {\it {\,d} x}\, {\it dy}. \]

490

Question 8.147

In Section 4.2, we saw that the length \(l({\bf c})\) of a curve \({\bf c}(t) = (x(t), y(t), z(t)), a\le t \le b\), was given by the formula \[ l({\bf c}) = \int d {\bf s} = \int_a^b \Big(\frac{{\,d} s}{{\it {\,d} t}}\Big) \,{\it {\,d} t} \] where, loosely speaking, \((d s)^2 = (d x)^2 + (d y)^2 + (d z)^2\), that is, \[ \frac{{\,d} s}{{\it {\,d} t}} = \sqrt{\Big(\frac{{\it {\,d} x}}{{\it {\,d} t}}\Big)^2 + \Big(\frac{{\it dy}}{{\it {\,d} t}}\Big)^2 + \Big(\frac{{\,d} z}{{\it {\,d} t}}\Big)^2}. \]

Now suppose a surface \(S\) is given in parametrized form by \({\Phi}(u,v) = (x(u,v),y(u,v), z(u,v))\), where \((u,v)\in D\). Show that the area of \(S\) can be expressed as \[ A(S) = \intop\!\!\!\intop\nolimits_{D}{\,d} S, \] where formally \((d S)^2= (d x \wedge {\it dy})^2 + (d y \wedge{\,d} z)^2 + (d z \wedge {\it {\,d} x})^2\), a formula requiring interpretation. [HINT: \[ {\it {\,d} x} = \frac{\partial x}{\partial u}{\,d} u + \frac{\partial x}{\partial v}{\,d} v, \] and similarly for \({\it dy}\) and \({\,d} z\). Use the law of forms for the basic 1-forms \({\,d} u\) and \({\,d} v\). Then \({\,d} S\) turns out to be a function times the basic 2-form \({\,d} u{\,d} v\), which we can integrate over \(D\).]