Can I think of a toy contour as any closed piecewise-smooth curve which is simple?












1












$begingroup$


enter image description here



enter image description here



enter image description here



enter image description here



enter image description here



I feel Stein doesn't give the specific definition of toy contour, can I think of a toy contour as any closed piecewise-smooth curve which is simple?










share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    enter image description here



    enter image description here



    enter image description here



    enter image description here



    enter image description here



    I feel Stein doesn't give the specific definition of toy contour, can I think of a toy contour as any closed piecewise-smooth curve which is simple?










    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      enter image description here



      enter image description here



      enter image description here



      enter image description here



      enter image description here



      I feel Stein doesn't give the specific definition of toy contour, can I think of a toy contour as any closed piecewise-smooth curve which is simple?










      share|cite|improve this question











      $endgroup$




      enter image description here



      enter image description here



      enter image description here



      enter image description here



      enter image description here



      I feel Stein doesn't give the specific definition of toy contour, can I think of a toy contour as any closed piecewise-smooth curve which is simple?







      complex-analysis analysis contour-integration






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 14 at 13:09







      Born to be proud

















      asked Jan 14 at 9:07









      Born to be proudBorn to be proud

      856510




      856510






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          Yes, that's a valid way to think about it.



          What's going on here is a theorem of topology:




          The Jordan Curve Theorem. If $C$ is any continuous simple closed curve then $mathbb R^2 - C$ has two connected components, one of which (called the "interior") is bounded and the other (called the "exterior") is unbounded.




          It's pretty clear from what the author is saying (in the first paragraph of the passages you have shown us) that he/she does not want to spend the time to either state or prove the Jordan Curve Theorem. This is certainly a reasonable decision for a textbook author, because it is a very hard theorem to prove.



          Instead, the words "toy contour" are used to mean any continuous simple closed curve for which the conclusion of the Jordan Curve Theorem is obvious. Examples of this would include any convex simple closed curve (an exercise). But he also wants some nonconvex examples such as the "keyhole" example, and the exercise is still pretty easy for those.



          And, by the way, there is another theorem with the same hypotheses but still stronger conclusions than the Jordan Curve Theorem, it is known as the Schönflies Theorem. The author's comments in that first paragraph lead me to think that the concept of "toy contour" might also be referring to those stronger conclusions in some form or another.






          share|cite|improve this answer











          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3073009%2fcan-i-think-of-a-toy-contour-as-any-closed-piecewise-smooth-curve-which-is-simpl%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2












            $begingroup$

            Yes, that's a valid way to think about it.



            What's going on here is a theorem of topology:




            The Jordan Curve Theorem. If $C$ is any continuous simple closed curve then $mathbb R^2 - C$ has two connected components, one of which (called the "interior") is bounded and the other (called the "exterior") is unbounded.




            It's pretty clear from what the author is saying (in the first paragraph of the passages you have shown us) that he/she does not want to spend the time to either state or prove the Jordan Curve Theorem. This is certainly a reasonable decision for a textbook author, because it is a very hard theorem to prove.



            Instead, the words "toy contour" are used to mean any continuous simple closed curve for which the conclusion of the Jordan Curve Theorem is obvious. Examples of this would include any convex simple closed curve (an exercise). But he also wants some nonconvex examples such as the "keyhole" example, and the exercise is still pretty easy for those.



            And, by the way, there is another theorem with the same hypotheses but still stronger conclusions than the Jordan Curve Theorem, it is known as the Schönflies Theorem. The author's comments in that first paragraph lead me to think that the concept of "toy contour" might also be referring to those stronger conclusions in some form or another.






            share|cite|improve this answer











            $endgroup$


















              2












              $begingroup$

              Yes, that's a valid way to think about it.



              What's going on here is a theorem of topology:




              The Jordan Curve Theorem. If $C$ is any continuous simple closed curve then $mathbb R^2 - C$ has two connected components, one of which (called the "interior") is bounded and the other (called the "exterior") is unbounded.




              It's pretty clear from what the author is saying (in the first paragraph of the passages you have shown us) that he/she does not want to spend the time to either state or prove the Jordan Curve Theorem. This is certainly a reasonable decision for a textbook author, because it is a very hard theorem to prove.



              Instead, the words "toy contour" are used to mean any continuous simple closed curve for which the conclusion of the Jordan Curve Theorem is obvious. Examples of this would include any convex simple closed curve (an exercise). But he also wants some nonconvex examples such as the "keyhole" example, and the exercise is still pretty easy for those.



              And, by the way, there is another theorem with the same hypotheses but still stronger conclusions than the Jordan Curve Theorem, it is known as the Schönflies Theorem. The author's comments in that first paragraph lead me to think that the concept of "toy contour" might also be referring to those stronger conclusions in some form or another.






              share|cite|improve this answer











              $endgroup$
















                2












                2








                2





                $begingroup$

                Yes, that's a valid way to think about it.



                What's going on here is a theorem of topology:




                The Jordan Curve Theorem. If $C$ is any continuous simple closed curve then $mathbb R^2 - C$ has two connected components, one of which (called the "interior") is bounded and the other (called the "exterior") is unbounded.




                It's pretty clear from what the author is saying (in the first paragraph of the passages you have shown us) that he/she does not want to spend the time to either state or prove the Jordan Curve Theorem. This is certainly a reasonable decision for a textbook author, because it is a very hard theorem to prove.



                Instead, the words "toy contour" are used to mean any continuous simple closed curve for which the conclusion of the Jordan Curve Theorem is obvious. Examples of this would include any convex simple closed curve (an exercise). But he also wants some nonconvex examples such as the "keyhole" example, and the exercise is still pretty easy for those.



                And, by the way, there is another theorem with the same hypotheses but still stronger conclusions than the Jordan Curve Theorem, it is known as the Schönflies Theorem. The author's comments in that first paragraph lead me to think that the concept of "toy contour" might also be referring to those stronger conclusions in some form or another.






                share|cite|improve this answer











                $endgroup$



                Yes, that's a valid way to think about it.



                What's going on here is a theorem of topology:




                The Jordan Curve Theorem. If $C$ is any continuous simple closed curve then $mathbb R^2 - C$ has two connected components, one of which (called the "interior") is bounded and the other (called the "exterior") is unbounded.




                It's pretty clear from what the author is saying (in the first paragraph of the passages you have shown us) that he/she does not want to spend the time to either state or prove the Jordan Curve Theorem. This is certainly a reasonable decision for a textbook author, because it is a very hard theorem to prove.



                Instead, the words "toy contour" are used to mean any continuous simple closed curve for which the conclusion of the Jordan Curve Theorem is obvious. Examples of this would include any convex simple closed curve (an exercise). But he also wants some nonconvex examples such as the "keyhole" example, and the exercise is still pretty easy for those.



                And, by the way, there is another theorem with the same hypotheses but still stronger conclusions than the Jordan Curve Theorem, it is known as the Schönflies Theorem. The author's comments in that first paragraph lead me to think that the concept of "toy contour" might also be referring to those stronger conclusions in some form or another.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Jan 14 at 15:17

























                answered Jan 14 at 15:09









                Lee MosherLee Mosher

                51k33889




                51k33889






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3073009%2fcan-i-think-of-a-toy-contour-as-any-closed-piecewise-smooth-curve-which-is-simpl%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Human spaceflight

                    Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

                    張江高科駅