The following is a plain TeX file containing all known errata to Differential Forms: A Complement to Vector Calculus.
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\centerline{Differential Forms: A Complement to Vector Calculus}
\centerline{Errata}
\pni
Page 3$~~~~~k$ or $\ell$ \hfill
\pni
Page 4$~~~~~$Ex. 2$~~~~~a)~3\varphi_3~-~4\varphi_4~~~~~b)~x \varphi_3~+~
y \varphi_4$ \hfill
\pni
$~~~~~~~~~~~~~~$Ex.5$~~~~~a)~x \psi_3~+~y \psi_4~~~~~b)~2y\psi_3~+~\psi_4$ \hfill
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$~~~~~~~~~~~~~~$Ex. 6$~~~~~d^{\prime})~\psi_2 \varphi_2$ \hfill
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Page 5$~~~~~$line 12$~~~~~(z~+~1) e^z dz$ \hfill
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$~~~~~~~~~~~~~~$line -1$~~~~~d(x^4~+~y^3~-~z^2) dz$ \hfill
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Page 10$~~~~~$line 15$~~~~~\psi~=~x^5 y^2 z^3$ \hfill
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Page 17$~~~~~$line -15$~~~~~$Let $\varphi~=$ \hfill
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$~~~~~~~~~~~~~~~$line -8$~~~~~{\partial \over {\partial y}}~ (x^2 y^3~+~x^4~+~c(y))$ \hfill
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Page 18$~~~~~$line 1$~~~~~$Let $\varphi~=$ \hfill
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$~~~~~~~~~~~~~~~$line 6$~~~~~$Then $x^2 z^3~+$ \hfill
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$~~~~~~~~~~~~~~~$line 7$~~~~~=~x^2 z^3 ~+~2xy~+$ \hfill
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$~~~~~~~~~~~~~~~$line -14$~~~~~x^2 yz^3~+~xy^2~+~4xz~+~2x~+~
2yz^3~-~y~-~2z^2~+~c$ \hfill
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Page 19$~~~~~$line -12$~~~~~\varphi~=$ \hfill
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Page 20$~~~~~$line 15$~~~~~$Let $\varphi~=$ \hfill
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Page 21$~~~~~$line 14$~~~~~$Let $\varphi~=$ \hfill
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$~~~~~~~~~~~~~~~$line 17$~~~~~$Then $\psi~=~(x^2 y^2 z^3~+~2x^3 y^3 z~-~x^4 y z^2)
dy dz$ \hfill
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Page 30$~~~~~$line 15 \hfill
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$~~~~~$DEFINITION 3.12: $(d?_1)^*~=~\varepsilon d?^1 ,~\varepsilon~=~\pm~1$, where
$(d?_1) (\varepsilon d?^1)~=~dx dy dz$. \hfill
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Page 31$~~~~~$line 13$~~~~~dx_1 , . . ., dx_n$ \hfill
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Page 32$~~~~~$Ex. 4 b)$~~~~~f~=~2xy^3$ \hfill
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Page 33$~~~~~$Ex. 7e)$~~~~~-4xy^2 z dx dy$ \hfill
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$~~~~~~~~~~~~~~~~$Ex. 10 \hfill
$$(d?_1)^*~=~\varepsilon d?^1~~{\rm where}~~(d?_1) (\varepsilon d?^1)~=~dx dy.$$
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Page 38$~~~~~$line 3$~~~~~$every \hfill
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Page 65$~~~~~$line 12$~~~~~C(x,y,0) dz.$ \hfill
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Page 67$~~~~~$line 22 \hfill
$$=~x_1 x_2 \varphi ({\bf i}, {\bf i})~+~x_1 y_2 \varphi ({\bf i}, {\bf j})~+~x_2 y_1
\varphi ({\bf j}, {\bf i})~+~y_1 y_2 \varphi ({\bf j}, {\bf j})$$
\pni
$~~~~~~~~~~~~~~~$line 24 \hfill
$$=~x_1 x_2 (0)~+~x_1 y_2 \varphi ({\bf i}, {\bf j})~+~x_2 y_1
(-\varphi ({\bf i}, {\bf j}))~+~y_1 y_2 (0)$$
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$~~~~~~~~~~~~~~~$line 25 \hfill
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$~~~~~~~~~~~~~~~~~~=~(x_1 y_2~-~x_2 y_1) \varphi ({\bf i}, {\bf j})$ \hfill
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Page 68$~~~~~$last line of footnote$~~~~~$in \hfill
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Page 71$~~~~~$line -10$~~~~~$point (2,5,-3) \hfill
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Page 75$~~~~~$line 7$~~~~~r (0)$ \hfill
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Page 76$~~~~~$line 3$~~~~~$definition 3.3 \hfill
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Page 82$~~~~~$line 9$~~~~~C(k(t)) h^{\prime} (t) dt.$ \hfill
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Page 91$~~~~~$line 14$~~~~~{\bf w}~=~k_*({\bf v})~=~(kr)^{\prime} (0)$ \hfill
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Page 94$~~~~~$lines 2,3,5,8$~~~~~${\bf w}$~~$should be$~~${\bf v} \hfill
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Page 97$~~~~~$line -9$~~~~~c_1 \varphi_1$ \hfill
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Page 101$~~~~~$line -4$~~~~~=~\int \limits_{I} A(f(t)) f^{\prime} (t) dt$ \hfill
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Page 102$~~~~~$line 4$~~~~~=\int \limits_{I} A(f(t)) f^{\prime} (t) dt$ \hfill
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Page 102$~~~~~$line 13$~~~~~\varphi~=$ \hfill
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$~~~~~~~~~~~~~~~~~$line 14$~~~~~(6t~+~2)^2$ \hfill
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$~~~~~~~~~~~~~~~~~$line 18$~~~~~\varphi~=$ \hfill
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$~~~~~~~~~~~~~~~~~$line -1$~~~~~\varphi~=$ \hfill
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Page 103$~~~~~$line 9$~~~~~\varphi~=$ \hfill
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$~~~~~~~~~~~~~~~~~$line -2$~~~~~\varphi^1~=$ \hfill
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Page 107$~~~~~$line -4$~~~~~on~C$ \hfill
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Page 109$~~~~~$line 4$~~~~~(A(r(t)), B(r(t)), C(r(t)))$ \hfill
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$~~~~~~~~~~~~~~~~~$line 6$~~~~\bigl.~+~C(f(t), g(t), h(t))\bigr ) dt$ \hfill
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Page 111$~~~~~$line -4$~~~~~\int \limits_{C_3} \varphi_1~=~16 \slash 15
~~~~~\int \limits_{C_3} \varphi_2~=~14 \slash 15$ \hfill
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Page 112$~~~~~$line 10$~~~~~16 \slash 15~+~14 \slash 15$ \hfill
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Page 112$~~~~~$line -6$~~~~~\partial C~=~\left\{ q \right\}~\cup~-\left\{ p \right\}$ \hfill
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Page 150$~~~~~$line -5 \hfill
$$dy~=~{{-4u v du~+~2(-v^2~+~u^2~+~1) dv} \over {(u^2~+~v^2~+~1)^2}}$$
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\eject
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Page 169:$~~$Replace lines -16 through -11 by: \hfill
\par
Since $k_2~=~k_1 \circ \ell,~k_2^*(\varphi)~=~\ell^*(k_1^*(\varphi))$ by proposition
III.3.61.
\par
For simplicity we complete the proof when $n~=~2$. As $k_1^*(\varphi)$ is a 2-form on
a region in $I\hskip-.13cmR^2$, we may write $k_1^* (\varphi)~=~f(x,y) dx dy$ for some
function $f(x,y)$. Then
$$\ell^* (k_1^* (\varphi))~=~\ell^* (f(x, y) dx dy)~=~f(\ell(u, v))
J (\ell) (u, v) du dv \eqno (5.17)$$
\pni
by proposition 3.34.
\par
Then the conclusion of the theorem becomes, by definition 3.2,
$$\pm~\int \limits_{T_1}~f(x,y) dA_{x y}~=~\pm~\int \limits_{T_2}~f(\ell(u,v))
J (\ell) (u,v) dA_{u v} . \eqno (5.18)$$
\par
We are assuming the orientations are all compatible. This implies that either
$T_1$ and $T_2$ both have the same orientation as surfaces in $I\hskip-.13cmR^2$,
so both sides of (5.18) get the same sign, and that $J (\ell) (u,v)$ is always positive,
or else $T_1$ and $T_2$ have opposite orientations as surfaces in $I\hskip-.13cmR^2$,
so the two sides of (5.18) have opposite signs, and that $J (\ell) (u,v)$ is always negative. In
either case, then, (5.18) is just the standard change-of-variable formula for double
integrals.
\par
The case of general $n$ is similar. (For $n~=~3$, use 4.8 and 4.2 instead of 3.34 and
3.2.) \hfill ${\,\lower0.9pt\vbox{\hrule \hbox{\vrule height 0.2 cm \hskip 0.2 cm \vrule
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\eject
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Page 190$~~~~~$Ex. 13$~~~~~$should be flush with left margin \hfill
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Page 194$~~~~~$footnote line 1$~~~~~$invertible \hfill
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Page 197$~~~~~$line 18 \hfill
$$a_2~\leq~x_2~\leq~b_2,~\ldots,~a_k~\leq~x_k~\leq~b_k$$
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Page 216$~~~~~$line -3$~~~~~$involve $dx_m$. \hfill
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Page 245$~~~~~$line -5$~~~~~$Jq$~~$should be$~~$Jg \hfill
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Page 247$~~~~~$I.1$~~~~~$1 a)$~~(4x^2~-~x)dx~+~3x dy$ \hfill
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$~~~~~~~~~~~~~~~~~~~~~~~~~~~~$b)$~~3x^2 dx~+~(-x^2~+~2xy~+~x~+~y) dy$ \hfill
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$~~~~~~~~~~~~~~~~~$I.1$~~~~~~$2 a)$~~(3x^3~-~4y^2 z) dx~+~(3yz~+~4xz) dy$ \hfill
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$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-(3x^2~+~3y^2~+~3z^2~+~8x~+~4)dz$ \hfill
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$~~~~~~~~~~~~~~~~~~~~~~~~~~~~$b)$~~(x^4~+~y^3 z) dx~+~(-x^3~-~xy^2~-~xz^2
~+~2xy ~+~y) dz$ \hfill
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Page 248$~~~~~$I.2$~~~~~$2 d)$~~xdy dz ~+~(y^2~-~2) dz dx~+~(-z~-~2yz)
dx dy$ \hfill
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Page 249$~~~~~$I.3$~~~~~$2 d)$~~x{\bf i}~+~(y^2~-~2) {\bf j}~+~(-z~-~2yz)
{\bf k}$ \hfill
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Page 251$~~~~~$III.1$~~~~~$2 e)$~~$34$~~~~~$2 f)$~~-17$ \hfill
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$~~~~~~~~~~~~~~~~~$III.2$~~~~~$2 d)$~~$27 \hfill
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$~~~~~~~~~~~~~~~~~~~~~~~~~~~$3 a)$~~$85 \hfill
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Page 252$~~~~~$IV.2$~~~~~$7 a)$~~x^3~-~2xy~+~2y^2$ \hfill
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Page 253$~~~~~$IV.4$~~~~~$2)$~~1 \slash 15$ \hfill
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Page 254$~~~~~$V.3$~~~~~$13)$~~-79 \slash 5$ \hfill
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$~~~~~~~~~~~~~~~~~$V.4$~~~~~$2)$~~1 \slash 15$ \hfill
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\bye