Questions on a Theorem and its proof in baby Rudin

**zzzhhh** · November 14th 2009, 03:45 AM

This is Theorem 10.27 in Page 267-268 of Rudin's "Principles of Mathematical Analysis", as the picture below shows.

I have three questions:
First, what is "oriented rectilinear $k$ -simplex"? Is it the same as oriented affine $k$ -simplex defined in the previous page?
Second, if $\sigma$ is in open set $E$ , that is, its range $\subseteq E$ , how to guarantee that $\bar\sigma$ is also in $E$ ? If not, integral $\int\nolimits_{\bar\sigma} \omega$ may not be meaningful at all.
Three, the proof deals only with the relation of Jacobians of $\sigma$ and $\bar\sigma$ , what about the coefficient function? In page 254, the integral is defined as $\int\nolimits_D \sum a_{i_1...i_k}(\sigma$ or $\bar\sigma(\mathbf u))J_{i_1...i_k}d\bf u$ , where $D$ is the parameter domain, but in general $\sigma(\mathbf u)\ne\bar\sigma(\mathbf u)$ , how can we say that the two integrals differ only in the sign?
If any other information is needed, please reply too. Thanks!

**HallsofIvy** · November 14th 2009, 05:18 AM

An "oriented rectilinear k-simplex" is a special case of an "oriented affine k-simplex" in that it requires that the boundaries be straight lines, planes, etc. in a given coordinate system which an oriented affine k-simplex does not.

Are you clear on the distinction between $\sigma$ and $\overline{\sigma}$ ? Unfortunately you didn't include the "(75)" where $\sigma$ was given but I think it is simply a simplex. $\overline{\sigma}$ , we are told, "is obtained from $\sigma$ by interchanging $p_0$ and $p_j$ ". That is, $\overline{\sigma}$ is exactly the same point set as $\sigma$ with possibly a different orientation. Being "in E" is a property of the point set, regardless of orientation.

**zzzhhh** · November 14th 2009, 07:57 AM

Thanks for the reply. The following pictures are other pages of the book containing definitions needed.
the following is the definition of "oriented affine $k$ -simplex"(Page 266):

This page contains the meaning of $\bar\sigma$ (Page 267):

This page defines the action of a $k$ -form on $k$ -surfaces(page 254), where $\frac{\partial(x_{i_1},...,x_{i_k})}{\partial(u_1, ...,u_k)}$ in (35) is just the $J_{i_1...i_k}$ I use in my first post.

Now, what is the formulation of "oriented rectilinear $k$ -simplex"? Is it all points $\mathbf x=(x_1,...,x_k)$ in $\mathbb R^k$ satisfying $\sum\limits_{i = 1}^k x_i=1$ and $x_i\geq 0$ for each $i$ ? Maybe its definition can explain why $\sigma(\mathbf u)=\bar\sigma(\mathbf u)$ .
Sorry for the missing parts of the book. If any other related information is needed, please reply, Thanks!

**zzzhhh** · November 20th 2009, 08:39 AM

The second question can be proved as follows:
It's enough to prove the situation where only one pair of vertices are interchanged, since general case is a series of compositions of this special case. The problem can be divided into two cases: a) $\mathbf p_0$ is not interchanged, b) $\mathbf p_0$ is interchanged. Suppose $\forall\mathbf u=(\alpha_1, ..., \alpha_k)\in Q^k$ . For the first case, let us suppose $\mathbf p_1$ and $\mathbf p_2$ are interchanged. Then $\bar\sigma(\mathbf u)=\sigma((\alpha_2,\alpha_1,\alpha_3,...,\alpha_k ))$ , where $(\alpha_2,\alpha_1,\alpha_3,...,\alpha_k)$ obviously belongs to $Q^k$ , so $\sigma((\alpha_2,\alpha_1,\alpha_3,...,\alpha_k))\ in E$ , that is, $\bar\sigma(\mathbf u)\in E$ . For the second case, without loss of generality, I suppose the pair $\mathbf p_0$ and $\mathbf p_1$ are exchanged. Then by (76) $\bar\sigma(\mathbf u)=\mathbf p_1+\alpha_1(\mathbf p_0-\mathbf p_1)+\sum\limits_{i = 2}^k\alpha_i(\mathbf p_i-\mathbf p_1)$ . Adding and subtracting $\mathbf p_0$ , we get $\mathbf p_0+(1-\alpha_1-\sum\limits_{i = 2}^k\alpha_i)(\mathbf p_1-\mathbf p_0)+\sum\limits_{i = 2}^k\alpha_i(\mathbf p_i-\mathbf p_0)$ . This is just the value of vector $(1-\alpha_1-\sum\limits_{i = 2}^k\alpha_i,\alpha_2,...,\alpha_k)$ under $\sigma$ . Noting that $0\leq 1-\alpha_1-\sum\limits_{i = 2}^k\alpha_i\leq 1$ since $\sum\limits_{i = 1}^k\alpha_i\leq 1$ , and that the sum of all components is $1-\alpha_1\leq 1$ , this vector is in $Q^k$ . So the above value $\in E$ , as desired. The same argument can be applied to prove that if $\sigma$ is in $E$ , then $\partial\sigma$ is also in $E$ .
I hope someone could help me with the other two questions: the formulation of "oriented rectilinear k-simplex" and why the two integrals are equal when considering the coefficient function. Thanks in advance!

**zzzhhh** · December 5th 2009, 03:53 PM

OK, I finally found the answer to the 3rd question:
the invariance of the value of the integral in spite of the fact that $\sigma(\mathbf u)\ne\bar\sigma(\mathbf u)$ and therefore $a_{i_1...i_k}(\sigma(\mathbf u))\ne a_{i_1...i_k}(\bar\sigma(\mathbf u))$ results from the theorem of change of variable. We have $\int_\sigma \omega=\int_{Q^k}\sum a_{i_1...i_k}(\sigma(\mathbf u))J_{i_1...i_k}(\mathbf u)d\bf u$ and $\int_{\bar\sigma}\omega=\int_{Q^k}\sum a_{i_1...i_k}(\bar\sigma(\mathbf u)){\bar J}_{i_1...i_k}(\mathbf u)d\bf u$ , where $J$ is the Jacobian generated from $\sigma$ and $\bar J$ is the Jacobian generated from $\bar\sigma$ . According to the proof given by the book, $\bar J(\mathbf u)=-J(\mathbf u)$ . As my last post indicated, $\bar\sigma(\mathbf u)=\sigma(B(\mathbf u))$ if $\mathbf p_0$ is not involved in the interchange, where $B$ is a flip defined in P249, or $\bar\sigma(\mathbf u)=\sigma(\bf G(u))$ otherwise, where $\bf G$ is a primitive defined in P248 with $m=1$ and $g(\mathbf u)=1-\alpha_1-\sum\limits_{i = 2}^k\alpha_i$ . For the former case, the differential of oriented affine simplex is just the linear transformation $A$ in (78) which is fixed for any $\mathbf u\in\mathbb R^k$ , so $-J(\mathbf u)=-J(B(\bf u))$ . Thus, $\int_{\bar\sigma}\omega=-\int_{Q^k}\sum a_{i_1...i_k}(\sigma(B(\mathbf u)))J_{i_1...i_k}(B(\mathbf u))d\bf u$ . Since $B$ merely interchanges two coordinates, it differs from $\int_\sigma \omega=\int_{Q^k}\sum a_{i_1...i_k}(\sigma(\mathbf u))J_{i_1...i_k}(\mathbf u)d\bf u$ only in the order in which $k$ integrations are carried out, but integral $\int_{Q^k}$ is independent of the integration order (See Example 10.4 in P247), then we get the equality. For the latter case, $\int_{\bar\sigma}\omega=-\int_{Q^k}\sum a_{i_1...i_k}(\sigma(\mathbf G(\mathbf u)))J_{i_1...i_k}(\mathbf G(\mathbf u))d\bf u$ . We try to prove ${\bar f}_{k-1} (\alpha_2,...,\alpha_k)=f_{k-1} (\alpha_2,...,\alpha_k)$ where ${\bar f}_{k-1} (\alpha_2,...,\alpha_k)=\int_0^1 \sum a_{i_1...i_k}(\sigma(\mathbf G(\mathbf u)))J_{i_1...i_k}(\mathbf G(\mathbf u))d\alpha_1$ and $f_{k-1} (\alpha_2,...,\alpha_k)=\int_0^1 \sum a_{i_1...i_k}(\sigma(\mathbf u))J_{i_1...i_k}(\mathbf u)d\alpha_1, 0\leq\alpha_i\leq 1, 1\leq i\leq k-1$ , because if this is the case, subsequent integral will lead to the same consequence $\int_{Q^k}$ as desired. If $\sum\limits_{i = 2}^k\alpha_i\geq 1$ , both integrand are 0 by the definition of $\int_{Q^k}$ and ${\bar f}_{k-1}=f_{k-1}$ trivially. If $0\leq\sum\limits_{i = 2}^k\alpha_i\leq 1$ , denoting $\sum\limits_{i = 2}^k\alpha_i$ as $S$ , $f_{k-1}=\int_0^{1-S}\sum a_{i_1...i_k}(\sigma(\alpha_1,\alpha_2,...,\alpha_ k))J_{i_1...i_k}(\alpha_1,\alpha_2,...,\alpha_k)d\ alpha_1$ because $\alpha_1\geq 1-S$ would get out of the standard simplex $Q^k$ and make the integrand zero. Since $\alpha_2,...,\alpha_k$ is fixed, $g(\mathbf u)=g(\alpha_1)=1-\alpha_1-S, g'(\alpha_1)=-1$ , this means $g$ is strictly decreasing, so $g(0)=1-S$ and $g(1-S)=0$ . In addition, $g$ is continuous on $[0,1-S]$ due to its differentiability. Applying the Theorem of change of variable (not the one in page 132-133 of this book because $g$ is not increasing, but the one in page 144-145 of Apostol's "Mathematical analysis") and Th6.17, we get $\int_0^{1-S}\sum a_{i_1...i_k}(\sigma(g(x_1),\alpha_2,...,\alpha_k) )J_{i_1...i_k}(g(x_1),\alpha_2,...,\alpha_k)d\alph a_1$ (note that $\int_{1-S}^0=-\int_0^{1-S}$ ). But for the same consideration of integral limits, ${\bar f}_{k-1}=\int_0^{1-S}\sum a_{i_1...i_k}(\sigma(\mathbf G(\mathbf u)))J_{i_1...i_k}(\mathbf G(\mathbf u))d\alpha_1$ which is just the above integral.
It is worth noting that I was not using the Theorem 10.9 of Change of variables in P252 because some conditions does not meet, but the idea in its proof, in particular its first paragraph, is almost the same as the above argument.
Since I have proved the general case of oriented affine $k$ -simplex, my first question on the definition of the so-called "oriented rectilinear $k$ -simplex" is no longer important (actually the exposition of the book that follows all uses the general oriented affine simplex case of the theorem).

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