Question 18.143

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 18.144

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 18.145

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 18.146

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 18.147

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 18.148

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 18.149

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 18.150

Using the differential form version of Stokes’ theorem, prove the vector field version in Section 18.2. Repeat for Gauss’ theorem.

Question 18.151

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

Question 18.152

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 18.153

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 18.154

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 18.155

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 18.156

In Section 12.3, 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, \[ \frac{{\,d} s}{{\it {\,d} t}} = \sqrt{\Big(\frac{{\it {\,d} x}}{{\it {\,d} t}}\Big)^2 + \Big(\frac{{\,d} y}{{\it {\,d} t}}\Big)^2 + \Big(\frac{{\,d} z}{{\it {\,d} t}}\Big)^2}. \]

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 \(D\).]