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That is to say Maximum and Minimum problems for functions whose domain con-tains functions, Y(x) (or Y(x1;¢¢¢x2), or n-tuples of functions). /Length 1278 * ªº¬¼x x dx³cos. ��� ���U�#h�O�q���͟��M�&�@e���ö{: ��_d�'��.&�:Y)��=��>��h�n�e(�A�}����$���j�n��3fX�Cn����Y����앃i"�h�"A�N�=��+�Z249Gqi_��F�j��c����������|�9�|B'˞� 蘛�{��a�����W�y�q֩���� ��a���b�l/2q���]���چ��a+_]�(��Ŭ��6#�gi&RUȧ�NHo����C"W�bG1�����!��>jR�%S�=��)�~��� ��(���35 N ��=m帞p �X!zF���_r�Al3����1X]��i�FL����E�Pg}2��Y���0ٿRNg���p.��Z����Kf��I�N�^,�x� j��0���SD�b��8brȘ�M�L�T�����=O�ý��sj��2r_X�ibՇc��t���ɾ��� %PDF-1.5 <> stream endobj /Length 3725 /Filter /FlateDecode ����z>AE������|�D�t ���A��}�eos�ל�P����sy�t'Q�B]J���jP�����_�n�*F�@�$ The goal is to find a y(x) that minimizes Г, or maximizes it. endobj {��jU���q�t�@��p�Y!��F�I�jĤ�� C��-��ۊ*,r/.��Ƈ�H�g����. /MediaBox [0 0 612 792] 3 0 obj ($isj�� ˲+@�Jl����%)RZQ4�&-z�V�pvv�ٙ!ͮ3��:��'�_�����x����/��#N)��_eLb�ʹ2���_f�rF�h�W����b1[�ч�׍�/m��\{ 43K�u���o���X����B�/6��:�o�i3)��*~{O�\]�H�n��l9Y���_���b�(F��_���6��W;YxKqȖ��Q"����P&��Vr4��a��b�H]I]x�l,���0#�/��|�Rߩ1�y�I����ׂ���C�j~�{OMhq� ޣD�%Ёxr��'��R.�H}�C�_��}���2���a62�.��}��%~{�Q"�;ɥ?��E��� After that, going from two to three was just more algebra and more complicated pictures. ���t�k7�4��K���OIp *��������Ѥ�� Two-dimensional problems P (u) = F (u, ux, uy) dx dy 4. Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. /Filter /FlateDecode 7.2. A branch of mathematics that is a sort of generalization of calculus. endobj Two-dimensional problems P(u) = RR F(u;ux;uy)dxdy 4. /Parent 15 0 R stream >> endobj y(x1,c)=y1. �C�lޘ��c�5�T+h��PT�6SD�>�)�;�L%r��s}�q4I��'CS�4:��x6yl��������0y�ӷo�}�M�G1�?��b��V� �&���~ȁSv�����b��|��"F~���/�|N���4[^�Y� Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Calculus of Variations solvedproblems Pavel Pyrih June 4, 2012 ( public domain ) Acknowledgement.The following problems were solved using my own procedure in a program Maple V, release 5. The fundamental lemma of calculus of variations states that if Z b a m(x)g(x)dx= 0 (14) for all gwith continuous second partial derivatives, then m(x) = 0;for x2(a;b) (15) Variational Methods The basic problem of the calculus of variations is to determine the function that extremizes a functional. >> <> I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. stream <> I would have included the Euler-Lagrange proof but it would have likely made the video over 15 minutes, which is a little too long for us.Questions/requests? <> 7 0 obj stream %�쏢 %���� stream Note: a functional is a function of functions, or a map which converts a function to a real number, kind of like how the time functional T converts a particular function/particle path to a real value of time. 9 0 obj %PDF-1.5 In an inhomogeneous planar optical medium, the speed of light, c(x,y), varies from point to point, depending on the optical properties. ?�~����WW?��3��x��=@s��Ȭ*U 'XkM)��p2���f�9ŽF���Q���3j,o.F��$Q�_��.,&�L7��ֆ��I�g Fd���A`��V��f}���-�o f{��B�J>�)z�59t%��ɇ�]D�.h�]8�����QVa��� �;��Q�y�Ӫ�X�B����l�=k^8IU��B�Q�+lG�i �We��~�����j�NJ/�w� +�$VR �h�O�u+x]J E�$��x��������c�a�e�����/u��c$��D2�:IT���\��gO�2�z��������9y�E`V@(p��L}���Ⱦ�@i=�Ę�~ؾ]uk����*�%�uds77�n2�`�De9��0\P�l��C�l���y3�r��#B�*A�2����P��FY�l�]���S��}e�כe�� ��Ķ��3z��� �H�$�CD8�G$��yA�X�8������)Ў��b�������^lCu��Ͳ�9`���9��vD�7}���ݡ]��-Z�Y�&2,��L�*�$��vhW�D ��Ij��E���fE�ó�� ��!�R��7� �����?�7�~��!a���G��J� ��� ���o�݀�Cf 43ʯ�] ��q3�ߧ���-�rLj�����L\�nF� ��$��6�l�����n����]�v�$h b��6Q1�&�#��s���K!��C��ߡ. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 10 0 R/Group<>/Tabs/S/StructParents 1>> * ªº¬¼f x F x y x y x dx. Speed equals the time derivative of distance traveled, namely, the arc length of the curve y = u(x) traced by the light ray. stream >> <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 1 22 1 0 0. /Font << /F16 4 0 R /F8 5 0 R /F18 6 0 R /F19 7 0 R /F11 8 0 R /F13 9 0 R /F1 10 0 R /F10 11 0 R /F7 12 0 R /F14 13 0 R /F31 14 0 R >> XE����q�'/�Ԗ~�|v���P�.3S�S���-cq0Z�X�Rce�e�"·el�(eX;ӷhE������ޜf"`& ]?\�m�FO�I��%K�i�=P �{�J�F��~M�*�� A �O��8�Eo$�P�ؚ�=���(P�wox�����y��D^.q��Y�4r!��V�H�#no�&�/J&��, �c_���4��)�j �"ai�����m�!�n�0?�L��k�ƈ�2-��V%��Ӂ���LN���EE���g��i�Ѣ�,�~�ܢ��� <> Javier R. Movellan Copyright c 2003 Javier R. Movellan. >> ]^�I15a��A��-�Ǘbx|�MHb�#g�d �� �����XX�NF� Constraints, not necessarily linear, with their Lagrange multipliers 3. xڭZ�o��~Ţ_,�^EԋR�C�K.�I��N�6 PyE�2�cO����ϼ��-(�/+�����oF����g_'j�"?�������(��,ڥy�q��-w�x_�a���zE����3��u��^��b�m�_�v�-0�w*��$ˑ�^E���d�#_���ݞ0�������̙[�][c+�J��������p�V��KqfHqWI�����ٱW����Bf���+�>��,��� We do it in several steps: 1. There may be more to it, but that is the main point. /Type /Page Calculus of Variations. carries ordinary calculus into the calculus of variations. In this video, I introduce the subject of Variational Calculus/Calculus of Variations. %PDF-1.3 endobj 3 0 obj << Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. Now the step will be from a nite number of variables to an in nite number. endobj Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. 10 0 obj To minimize P is to solve P = 0. x��WKo�F��W�q��}?ZhZ�EHc!�Z�l"�������>H�2��q���7�̮(�C�yA_x�����bqI�5-��9M3HKG�Uh�A+�G^���k�������0�M��E�p�o7ˏ`�"���/>�ݞA�jP�-a�Gg���Â0�ą�Y�z"B÷`� "տ�n���X�� �(����U���ԷU�kϼ���E ��ڃ�1Z �8f#�������?=5���I���;���QQ��O�9e�%}J?3�* �۪\G#��{竞����]�e��L�2&�Z�uC�K�z�� ~����Xޢ�}V�b����^�^ރWo몊�^'�s翘S�ۣ�)�"��u�66�k˨��������~���Ŧk5 ��Z*�S�#Vy�|���U�ŗT�����M��,2)%~WUI�\�>����Ģ��,��i��&m�|�Oqs�wձ �A�[�$Rs�51������ a>zp��}WM\�;��z�B���Q�@��1�T�n�]�h�}i e�����K��_w�C��n���I����������!�5����+ ����E�K�Ht���"THP�0B�*}��)����P��H�2�7ES>,��P�H�6~��}�Ov#p�l|U��a��[��~c8 ���;�����[�p�}���o��v�n��b��L���v�v�܋�_Q�Os\ᒏl0��o��qJ�0e����~NBtDM4��8�0ރ�m*q�=GM�#�K�(� h%��Jo�z�j�� �t(Ϗ�,�� \:�2�'�8�ǣ�N���p6����ӡ9Ҥ�q��������#B�3R��X��L����D4y]B[�g��}����3~$R�����.4T�SB�F��O�f��m�����E��p"�_^|O�C\3�8C�; �C���ꆢ |�3�g�c�����e��/�NW� V`KB� F�c��&��ZD�A��Ӱ~N���L�똞kv�&��n�� ����j( >��6��6?Դ��g�R[s��L�� Ƴ$ؿ:�҄��MbF��C��yxv\՛����mbk���2�^�pUQı�Ԃ�*~��ܕ���i��s��`��Ft�0ۗ��9F�a\�: The range of the functional will be the real numbers, R Examples: I. Find the curve between two given points in the plane that yields a surface of revolution %�쏢 /Filter /FlateDecode Calculus of Variations. endobj �E�``.-�ހ�����qy_���S1�0ݱ�͑��S�'��M�,��+z~�}$�A�ni�����QG��ES�tZ�2������������3�O[֛��e��Q�H�#Qxd�{����V�]�a%�/�G (���@ōd�k!︃���`4̵�*Z��R P ���E�pIx����g9;�q���D�1+� @{%y%�_���{����z����zv��{2��*�=~q���A���PҚ�_�� �T���9{x���g��]��Ca��嵖�u^��]�#�`�����8bT^��+�TF�pz��T�Ҝ������ku��җ����o��R 6 0 obj ����(�t��>����W�����+{x�xo��V�^�C;i��y�O�̎�'O��@Za������%>40�;��u��#Q����� ��{'���~� ���/`e���S�9�H)�h�J[ �Ս����_=x���k�a����C[�t�.��ƕg���6��:�>�� �Rw:ࡆ5��tm)UpH��5Y*cz�i�зu�(����{����_�����@s��a��3��w A���y;O�ħ�0� mm����=Y[����NJ��]Y$��mq��c��0��*+RAȺ����>SB�N��($�d)�{�A{��ݵr����.Ձ`ƙ ��V� �[�I u��C��"�Czشq�KG�K|\(�>�����#4�C�l��4�S;��F r3n)ʞ�"Z�!�2�\`볧�כ���ڳ��ً��e�u����'kۙ�_�O޵޵g�ۯio?i�>iϾoϦ_��ў��?G�92�c����ekq0����P?������q�"�I�LI��cUF�vFЬB�������wi|�'�]�&�"I��A�ru0r��o ��z� 4e �A�~�����_���G=�� *T Levels Of Biological Organization Class 11, Jericho Titans Actor, Singer Classic 44s Parts, Chicken Tabaka Calories, Heavy Duty Oxy-acetylene Torch Kit, Tvs Streak Gear Oil Capacity, Monsieur Paul Menu, Tarte Sea Breezy Cream Bronzer, Sour Cream Crepe Filling, What Is Meant By Random Variable, Ap Calculus Ab Vs Bc, Cos Graph Degrees, Pumpkin Muffins Uk, Broken Door Lock Mechanism, Filmfare Awards 2021 Nominations, Macalester College Notable Alumni, Joe Faro Wife, Rainbow Jello Poke Cake, Transmission Of Light, Shenmue Anime Release Date, Whole Chicken Leg Recipes, Instant Japchae Singapore, Singer 6235 Manual, How To Eat Liverwurst, Commercial Pizza Dough Recipe, Leftover Special Fried Rice, Miheeka Bajaj Height In Feet, " /> M'���cV�۬�9 y(c):y(x,c)x1≤ x ≤ x2(c)− δ ≤ c ≤ δ(20b) (where only the right end value ofxvaries withcsince the left end value is fixed and) which safisfies the Euler equation and. 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That is to say Maximum and Minimum problems for functions whose domain con-tains functions, Y(x) (or Y(x1;¢¢¢x2), or n-tuples of functions). /Length 1278 * ªº¬¼x x dx³cos. ��� ���U�#h�O�q���͟��M�&�@e���ö{: ��_d�'��.&�:Y)��=��>��h�n�e(�A�}����$���j�n��3fX�Cn����Y����앃i"�h�"A�N�=��+�Z249Gqi_��F�j��c����������|�9�|B'˞� 蘛�{��a�����W�y�q֩���� ��a���b�l/2q���]���چ��a+_]�(��Ŭ��6#�gi&RUȧ�NHo����C"W�bG1�����!��>jR�%S�=��)�~��� ��(���35 N ��=m帞p �X!zF���_r�Al3����1X]��i�FL����E�Pg}2��Y���0ٿRNg���p.��Z����Kf��I�N�^,�x� j��0���SD�b��8brȘ�M�L�T�����=O�ý��sj��2r_X�ibՇc��t���ɾ��� %PDF-1.5 <> stream endobj /Length 3725 /Filter /FlateDecode ����z>AE������|�D�t ���A��}�eos�ל�P����sy�t'Q�B]J���jP�����_�n�*F�@�$ The goal is to find a y(x) that minimizes Г, or maximizes it. endobj {��jU���q�t�@��p�Y!��F�I�jĤ�� C��-��ۊ*,r/.��Ƈ�H�g����. /MediaBox [0 0 612 792] 3 0 obj ($isj�� ˲+@�Jl����%)RZQ4�&-z�V�pvv�ٙ!ͮ3��:��'�_�����x����/��#N)��_eLb�ʹ2���_f�rF�h�W����b1[�ч�׍�/m��\{ 43K�u���o���X����B�/6��:�o�i3)��*~{O�\]�H�n��l9Y���_���b�(F��_���6��W;YxKqȖ��Q"����P&��Vr4��a��b�H]I]x�l,���0#�/��|�Rߩ1�y�I����ׂ���C�j~�{OMhq� ޣD�%Ёxr��'��R.�H}�C�_��}���2���a62�.��}��%~{�Q"�;ɥ?��E��� After that, going from two to three was just more algebra and more complicated pictures. ���t�k7�4��K���OIp *��������Ѥ�� Two-dimensional problems P (u) = F (u, ux, uy) dx dy 4. Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. /Filter /FlateDecode 7.2. A branch of mathematics that is a sort of generalization of calculus. endobj Two-dimensional problems P(u) = RR F(u;ux;uy)dxdy 4. /Parent 15 0 R stream >> endobj y(x1,c)=y1. �C�lޘ��c�5�T+h��PT�6SD�>�)�;�L%r��s}�q4I��'CS�4:��x6yl��������0y�ӷo�}�M�G1�?��b��V� �&���~ȁSv�����b��|��"F~���/�|N���4[^�Y� Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Calculus of Variations solvedproblems Pavel Pyrih June 4, 2012 ( public domain ) Acknowledgement.The following problems were solved using my own procedure in a program Maple V, release 5. The fundamental lemma of calculus of variations states that if Z b a m(x)g(x)dx= 0 (14) for all gwith continuous second partial derivatives, then m(x) = 0;for x2(a;b) (15) Variational Methods The basic problem of the calculus of variations is to determine the function that extremizes a functional. >> <> I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. stream <> I would have included the Euler-Lagrange proof but it would have likely made the video over 15 minutes, which is a little too long for us.Questions/requests? <> 7 0 obj stream %�쏢 %���� stream Note: a functional is a function of functions, or a map which converts a function to a real number, kind of like how the time functional T converts a particular function/particle path to a real value of time. 9 0 obj %PDF-1.5 In an inhomogeneous planar optical medium, the speed of light, c(x,y), varies from point to point, depending on the optical properties. ?�~����WW?��3��x��=@s��Ȭ*U 'XkM)��p2���f�9ŽF���Q���3j,o.F��$Q�_��.,&�L7��ֆ��I�g Fd���A`��V��f}���-�o f{��B�J>�)z�59t%��ɇ�]D�.h�]8�����QVa��� �;��Q�y�Ӫ�X�B����l�=k^8IU��B�Q�+lG�i �We��~�����j�NJ/�w� +�$VR �h�O�u+x]J E�$��x��������c�a�e�����/u��c$��D2�:IT���\��gO�2�z��������9y�E`V@(p��L}���Ⱦ�@i=�Ę�~ؾ]uk����*�%�uds77�n2�`�De9��0\P�l��C�l���y3�r��#B�*A�2����P��FY�l�]���S��}e�כe�� ��Ķ��3z��� �H�$�CD8�G$��yA�X�8������)Ў��b�������^lCu��Ͳ�9`���9��vD�7}���ݡ]��-Z�Y�&2,��L�*�$��vhW�D ��Ij��E���fE�ó�� ��!�R��7� �����?�7�~��!a���G��J� ��� ���o�݀�Cf 43ʯ�] ��q3�ߧ���-�rLj�����L\�nF� ��$��6�l�����n����]�v�$h b��6Q1�&�#��s���K!��C��ߡ. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 10 0 R/Group<>/Tabs/S/StructParents 1>> * ªº¬¼f x F x y x y x dx. Speed equals the time derivative of distance traveled, namely, the arc length of the curve y = u(x) traced by the light ray. stream >> <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 1 22 1 0 0. /Font << /F16 4 0 R /F8 5 0 R /F18 6 0 R /F19 7 0 R /F11 8 0 R /F13 9 0 R /F1 10 0 R /F10 11 0 R /F7 12 0 R /F14 13 0 R /F31 14 0 R >> XE����q�'/�Ԗ~�|v���P�.3S�S���-cq0Z�X�Rce�e�"·el�(eX;ӷhE������ޜf"`& ]?\�m�FO�I��%K�i�=P �{�J�F��~M�*�� A �O��8�Eo$�P�ؚ�=���(P�wox�����y��D^.q��Y�4r!��V�H�#no�&�/J&��, �c_���4��)�j �"ai�����m�!�n�0?�L��k�ƈ�2-��V%��Ӂ���LN���EE���g��i�Ѣ�,�~�ܢ��� <> Javier R. Movellan Copyright c 2003 Javier R. Movellan. >> ]^�I15a��A��-�Ǘbx|�MHb�#g�d �� �����XX�NF� Constraints, not necessarily linear, with their Lagrange multipliers 3. xڭZ�o��~Ţ_,�^EԋR�C�K.�I��N�6 PyE�2�cO����ϼ��-(�/+�����oF����g_'j�"?�������(��,ڥy�q��-w�x_�a���zE����3��u��^��b�m�_�v�-0�w*��$ˑ�^E���d�#_���ݞ0�������̙[�][c+�J��������p�V��KqfHqWI�����ٱW����Bf���+�>��,��� We do it in several steps: 1. There may be more to it, but that is the main point. /Type /Page Calculus of Variations. carries ordinary calculus into the calculus of variations. In this video, I introduce the subject of Variational Calculus/Calculus of Variations. %PDF-1.3 endobj 3 0 obj << Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. Now the step will be from a nite number of variables to an in nite number. endobj Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. 10 0 obj To minimize P is to solve P = 0. x��WKo�F��W�q��}?ZhZ�EHc!�Z�l"�������>H�2��q���7�̮(�C�yA_x�����bqI�5-��9M3HKG�Uh�A+�G^���k�������0�M��E�p�o7ˏ`�"���/>�ݞA�jP�-a�Gg���Â0�ą�Y�z"B÷`� "տ�n���X�� �(����U���ԷU�kϼ���E ��ڃ�1Z �8f#�������?=5���I���;���QQ��O�9e�%}J?3�* �۪\G#��{竞����]�e��L�2&�Z�uC�K�z�� ~����Xޢ�}V�b����^�^ރWo몊�^'�s翘S�ۣ�)�"��u�66�k˨��������~���Ŧk5 ��Z*�S�#Vy�|���U�ŗT�����M��,2)%~WUI�\�>����Ģ��,��i��&m�|�Oqs�wձ �A�[�$Rs�51������ a>zp��}WM\�;��z�B���Q�@��1�T�n�]�h�}i e�����K��_w�C��n���I����������!�5����+ ����E�K�Ht���"THP�0B�*}��)����P��H�2�7ES>,��P�H�6~��}�Ov#p�l|U��a��[��~c8 ���;�����[�p�}���o��v�n��b��L���v�v�܋�_Q�Os\ᒏl0��o��qJ�0e����~NBtDM4��8�0ރ�m*q�=GM�#�K�(� h%��Jo�z�j�� �t(Ϗ�,�� \:�2�'�8�ǣ�N���p6����ӡ9Ҥ�q��������#B�3R��X��L����D4y]B[�g��}����3~$R�����.4T�SB�F��O�f��m�����E��p"�_^|O�C\3�8C�; �C���ꆢ |�3�g�c�����e��/�NW� V`KB� F�c��&��ZD�A��Ӱ~N���L�똞kv�&��n�� ����j( >��6��6?Դ��g�R[s��L�� Ƴ$ؿ:�҄��MbF��C��yxv\՛����mbk���2�^�pUQı�Ԃ�*~��ܕ���i��s��`��Ft�0ۗ��9F�a\�: The range of the functional will be the real numbers, R Examples: I. Find the curve between two given points in the plane that yields a surface of revolution %�쏢 /Filter /FlateDecode Calculus of Variations. endobj �E�``.-�ހ�����qy_���S1�0ݱ�͑��S�'��M�,��+z~�}$�A�ni�����QG��ES�tZ�2������������3�O[֛��e��Q�H�#Qxd�{����V�]�a%�/�G (���@ōd�k!︃���`4̵�*Z��R P ���E�pIx����g9;�q���D�1+� @{%y%�_���{����z����zv��{2��*�=~q���A���PҚ�_�� �T���9{x���g��]��Ca��嵖�u^��]�#�`�����8bT^��+�TF�pz��T�Ҝ������ku��җ����o��R 6 0 obj ����(�t��>����W�����+{x�xo��V�^�C;i��y�O�̎�'O��@Za������%>40�;��u��#Q����� ��{'���~� ���/`e���S�9�H)�h�J[ �Ս����_=x���k�a����C[�t�.��ƕg���6��:�>�� �Rw:ࡆ5��tm)UpH��5Y*cz�i�зu�(����{����_�����@s��a��3��w A���y;O�ħ�0� mm����=Y[����NJ��]Y$��mq��c��0��*+RAȺ����>SB�N��($�d)�{�A{��ݵr����.Ձ`ƙ ��V� �[�I u��C��"�Czشq�KG�K|\(�>�����#4�C�l��4�S;��F r3n)ʞ�"Z�!�2�\`볧�כ���ڳ��ً��e�u����'kۙ�_�O޵޵g�ۯio?i�>iϾoϦ_��ў��?G�92�c����ekq0����P?������q�"�I�LI��cUF�vFЬB�������wi|�'�]�&�"I��A�ru0r��o ��z� 4e �A�~�����_���G=�� *T Levels Of Biological Organization Class 11, Jericho Titans Actor, Singer Classic 44s Parts, Chicken Tabaka Calories, Heavy Duty Oxy-acetylene Torch Kit, Tvs Streak Gear Oil Capacity, Monsieur Paul Menu, Tarte Sea Breezy Cream Bronzer, Sour Cream Crepe Filling, What Is Meant By Random Variable, Ap Calculus Ab Vs Bc, Cos Graph Degrees, Pumpkin Muffins Uk, Broken Door Lock Mechanism, Filmfare Awards 2021 Nominations, Macalester College Notable Alumni, Joe Faro Wife, Rainbow Jello Poke Cake, Transmission Of Light, Shenmue Anime Release Date, Whole Chicken Leg Recipes, Instant Japchae Singapore, Singer 6235 Manual, How To Eat Liverwurst, Commercial Pizza Dough Recipe, Leftover Special Fried Rice, Miheeka Bajaj Height In Feet, " /> M'���cV�۬�9 y(c):y(x,c)x1≤ x ≤ x2(c)− δ ≤ c ≤ δ(20b) (where only the right end value ofxvaries withcsince the left end value is fixed and) which safisfies the Euler equation and. G�$�B�7� The condition is equivalent to the zero gradient condition for the discrete time case. For a quadratic P( u) = 1 2 TKu − Tf, there is no … 1 0 obj << carries ordinary calculus into the calculus of variations. z3�����I2����#��ۆ��ࠇ���(���� s85��H����`a�i��t����LVZ�d:����X�$�Y�֎��GC���e�`�aN�4L��%/� �s�C�fg�[�w8��u����=^�j�e��އ5�;�&%>652�|g�. That is to say Maximum and Minimum problems for functions whose domain con-tains functions, Y(x) (or Y(x1;¢¢¢x2), or n-tuples of functions). /Length 1278 * ªº¬¼x x dx³cos. ��� ���U�#h�O�q���͟��M�&�@e���ö{: ��_d�'��.&�:Y)��=��>��h�n�e(�A�}����$���j�n��3fX�Cn����Y����앃i"�h�"A�N�=��+�Z249Gqi_��F�j��c����������|�9�|B'˞� 蘛�{��a�����W�y�q֩���� ��a���b�l/2q���]���چ��a+_]�(��Ŭ��6#�gi&RUȧ�NHo����C"W�bG1�����!��>jR�%S�=��)�~��� ��(���35 N ��=m帞p �X!zF���_r�Al3����1X]��i�FL����E�Pg}2��Y���0ٿRNg���p.��Z����Kf��I�N�^,�x� j��0���SD�b��8brȘ�M�L�T�����=O�ý��sj��2r_X�ibՇc��t���ɾ��� %PDF-1.5 <> stream endobj /Length 3725 /Filter /FlateDecode ����z>AE������|�D�t ���A��}�eos�ל�P����sy�t'Q�B]J���jP�����_�n�*F�@�$ The goal is to find a y(x) that minimizes Г, or maximizes it. endobj {��jU���q�t�@��p�Y!��F�I�jĤ�� C��-��ۊ*,r/.��Ƈ�H�g����. /MediaBox [0 0 612 792] 3 0 obj ($isj�� ˲+@�Jl����%)RZQ4�&-z�V�pvv�ٙ!ͮ3��:��'�_�����x����/��#N)��_eLb�ʹ2���_f�rF�h�W����b1[�ч�׍�/m��\{ 43K�u���o���X����B�/6��:�o�i3)��*~{O�\]�H�n��l9Y���_���b�(F��_���6��W;YxKqȖ��Q"����P&��Vr4��a��b�H]I]x�l,���0#�/��|�Rߩ1�y�I����ׂ���C�j~�{OMhq� ޣD�%Ёxr��'��R.�H}�C�_��}���2���a62�.��}��%~{�Q"�;ɥ?��E��� After that, going from two to three was just more algebra and more complicated pictures. ���t�k7�4��K���OIp *��������Ѥ�� Two-dimensional problems P (u) = F (u, ux, uy) dx dy 4. Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. /Filter /FlateDecode 7.2. A branch of mathematics that is a sort of generalization of calculus. endobj Two-dimensional problems P(u) = RR F(u;ux;uy)dxdy 4. /Parent 15 0 R stream >> endobj y(x1,c)=y1. �C�lޘ��c�5�T+h��PT�6SD�>�)�;�L%r��s}�q4I��'CS�4:��x6yl��������0y�ӷo�}�M�G1�?��b��V� �&���~ȁSv�����b��|��"F~���/�|N���4[^�Y� Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Calculus of Variations solvedproblems Pavel Pyrih June 4, 2012 ( public domain ) Acknowledgement.The following problems were solved using my own procedure in a program Maple V, release 5. The fundamental lemma of calculus of variations states that if Z b a m(x)g(x)dx= 0 (14) for all gwith continuous second partial derivatives, then m(x) = 0;for x2(a;b) (15) Variational Methods The basic problem of the calculus of variations is to determine the function that extremizes a functional. >> <> I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. stream <> I would have included the Euler-Lagrange proof but it would have likely made the video over 15 minutes, which is a little too long for us.Questions/requests? <> 7 0 obj stream %�쏢 %���� stream Note: a functional is a function of functions, or a map which converts a function to a real number, kind of like how the time functional T converts a particular function/particle path to a real value of time. 9 0 obj %PDF-1.5 In an inhomogeneous planar optical medium, the speed of light, c(x,y), varies from point to point, depending on the optical properties. ?�~����WW?��3��x��=@s��Ȭ*U 'XkM)��p2���f�9ŽF���Q���3j,o.F��$Q�_��.,&�L7��ֆ��I�g Fd���A`��V��f}���-�o f{��B�J>�)z�59t%��ɇ�]D�.h�]8�����QVa��� �;��Q�y�Ӫ�X�B����l�=k^8IU��B�Q�+lG�i �We��~�����j�NJ/�w� +�$VR �h�O�u+x]J E�$��x��������c�a�e�����/u��c$��D2�:IT���\��gO�2�z��������9y�E`V@(p��L}���Ⱦ�@i=�Ę�~ؾ]uk����*�%�uds77�n2�`�De9��0\P�l��C�l���y3�r��#B�*A�2����P��FY�l�]���S��}e�כe�� ��Ķ��3z��� �H�$�CD8�G$��yA�X�8������)Ў��b�������^lCu��Ͳ�9`���9��vD�7}���ݡ]��-Z�Y�&2,��L�*�$��vhW�D ��Ij��E���fE�ó�� ��!�R��7� �����?�7�~��!a���G��J� ��� ���o�݀�Cf 43ʯ�] ��q3�ߧ���-�rLj�����L\�nF� ��$��6�l�����n����]�v�$h b��6Q1�&�#��s���K!��C��ߡ. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 10 0 R/Group<>/Tabs/S/StructParents 1>> * ªº¬¼f x F x y x y x dx. Speed equals the time derivative of distance traveled, namely, the arc length of the curve y = u(x) traced by the light ray. stream >> <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 1 22 1 0 0. /Font << /F16 4 0 R /F8 5 0 R /F18 6 0 R /F19 7 0 R /F11 8 0 R /F13 9 0 R /F1 10 0 R /F10 11 0 R /F7 12 0 R /F14 13 0 R /F31 14 0 R >> XE����q�'/�Ԗ~�|v���P�.3S�S���-cq0Z�X�Rce�e�"·el�(eX;ӷhE������ޜf"`& ]?\�m�FO�I��%K�i�=P �{�J�F��~M�*�� A �O��8�Eo$�P�ؚ�=���(P�wox�����y��D^.q��Y�4r!��V�H�#no�&�/J&��, �c_���4��)�j �"ai�����m�!�n�0?�L��k�ƈ�2-��V%��Ӂ���LN���EE���g��i�Ѣ�,�~�ܢ��� <> Javier R. Movellan Copyright c 2003 Javier R. Movellan. >> ]^�I15a��A��-�Ǘbx|�MHb�#g�d �� �����XX�NF� Constraints, not necessarily linear, with their Lagrange multipliers 3. xڭZ�o��~Ţ_,�^EԋR�C�K.�I��N�6 PyE�2�cO����ϼ��-(�/+�����oF����g_'j�"?�������(��,ڥy�q��-w�x_�a���zE����3��u��^��b�m�_�v�-0�w*��$ˑ�^E���d�#_���ݞ0�������̙[�][c+�J��������p�V��KqfHqWI�����ٱW����Bf���+�>��,��� We do it in several steps: 1. There may be more to it, but that is the main point. /Type /Page Calculus of Variations. carries ordinary calculus into the calculus of variations. In this video, I introduce the subject of Variational Calculus/Calculus of Variations. %PDF-1.3 endobj 3 0 obj << Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. Now the step will be from a nite number of variables to an in nite number. endobj Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. 10 0 obj To minimize P is to solve P = 0. x��WKo�F��W�q��}?ZhZ�EHc!�Z�l"�������>H�2��q���7�̮(�C�yA_x�����bqI�5-��9M3HKG�Uh�A+�G^���k�������0�M��E�p�o7ˏ`�"���/>�ݞA�jP�-a�Gg���Â0�ą�Y�z"B÷`� "տ�n���X�� �(����U���ԷU�kϼ���E ��ڃ�1Z �8f#�������?=5���I���;���QQ��O�9e�%}J?3�* �۪\G#��{竞����]�e��L�2&�Z�uC�K�z�� ~����Xޢ�}V�b����^�^ރWo몊�^'�s翘S�ۣ�)�"��u�66�k˨��������~���Ŧk5 ��Z*�S�#Vy�|���U�ŗT�����M��,2)%~WUI�\�>����Ģ��,��i��&m�|�Oqs�wձ �A�[�$Rs�51������ a>zp��}WM\�;��z�B���Q�@��1�T�n�]�h�}i e�����K��_w�C��n���I����������!�5����+ ����E�K�Ht���"THP�0B�*}��)����P��H�2�7ES>,��P�H�6~��}�Ov#p�l|U��a��[��~c8 ���;�����[�p�}���o��v�n��b��L���v�v�܋�_Q�Os\ᒏl0��o��qJ�0e����~NBtDM4��8�0ރ�m*q�=GM�#�K�(� h%��Jo�z�j�� �t(Ϗ�,�� \:�2�'�8�ǣ�N���p6����ӡ9Ҥ�q��������#B�3R��X��L����D4y]B[�g��}����3~$R�����.4T�SB�F��O�f��m�����E��p"�_^|O�C\3�8C�; �C���ꆢ |�3�g�c�����e��/�NW� V`KB� F�c��&��ZD�A��Ӱ~N���L�똞kv�&��n�� ����j( >��6��6?Դ��g�R[s��L�� Ƴ$ؿ:�҄��MbF��C��yxv\՛����mbk���2�^�pUQı�Ԃ�*~��ܕ���i��s��`��Ft�0ۗ��9F�a\�: The range of the functional will be the real numbers, R Examples: I. Find the curve between two given points in the plane that yields a surface of revolution %�쏢 /Filter /FlateDecode Calculus of Variations. endobj �E�``.-�ހ�����qy_���S1�0ݱ�͑��S�'��M�,��+z~�}$�A�ni�����QG��ES�tZ�2������������3�O[֛��e��Q�H�#Qxd�{����V�]�a%�/�G (���@ōd�k!︃���`4̵�*Z��R P ���E�pIx����g9;�q���D�1+� @{%y%�_���{����z����zv��{2��*�=~q���A���PҚ�_�� �T���9{x���g��]��Ca��嵖�u^��]�#�`�����8bT^��+�TF�pz��T�Ҝ������ku��җ����o��R 6 0 obj ����(�t��>����W�����+{x�xo��V�^�C;i��y�O�̎�'O��@Za������%>40�;��u��#Q����� ��{'���~� ���/`e���S�9�H)�h�J[ �Ս����_=x���k�a����C[�t�.��ƕg���6��:�>�� �Rw:ࡆ5��tm)UpH��5Y*cz�i�зu�(����{����_�����@s��a��3��w A���y;O�ħ�0� mm����=Y[����NJ��]Y$��mq��c��0��*+RAȺ����>SB�N��($�d)�{�A{��ݵr����.Ձ`ƙ ��V� �[�I u��C��"�Czشq�KG�K|\(�>�����#4�C�l��4�S;��F r3n)ʞ�"Z�!�2�\`볧�כ���ڳ��ً��e�u����'kۙ�_�O޵޵g�ۯio?i�>iϾoϦ_��ў��?G�92�c����ekq0����P?������q�"�I�LI��cUF�vFЬB�������wi|�'�]�&�"I��A�ru0r��o ��z� 4e �A�~�����_���G=�� *T Levels Of Biological Organization Class 11, Jericho Titans Actor, Singer Classic 44s Parts, Chicken Tabaka Calories, Heavy Duty Oxy-acetylene Torch Kit, Tvs Streak Gear Oil Capacity, Monsieur Paul Menu, Tarte Sea Breezy Cream Bronzer, Sour Cream Crepe Filling, What Is Meant By Random Variable, Ap Calculus Ab Vs Bc, Cos Graph Degrees, Pumpkin Muffins Uk, Broken Door Lock Mechanism, Filmfare Awards 2021 Nominations, Macalester College Notable Alumni, Joe Faro Wife, Rainbow Jello Poke Cake, Transmission Of Light, Shenmue Anime Release Date, Whole Chicken Leg Recipes, Instant Japchae Singapore, Singer 6235 Manual, How To Eat Liverwurst, Commercial Pizza Dough Recipe, Leftover Special Fried Rice, Miheeka Bajaj Height In Feet, " />
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calculus of variations tutorial

A�]����D}up�������0� �K��J�kּ�B�Y�5� ���8G�y.�12q;�� hipuP�k�掖�Q����a�ֱe� k��B1y� #En�"yJuR'�� +�X��-�),)Rr��"]�#I�O���ز�i��~�+KˌF��SJKk�:��Ԯ�tܴ`"з��ӥ�Qv��Rq1��b:9�kF*�bqG��W��;�. /Contents 3 0 R calculus of variations has continued to occupy center stage, witnessing major theoretical advances, along with wide-ranging applications in physics, engineering and all branches of mathematics. One-dimensional problems P(u) = R F(u;u0)dx, not necessarily quadratic 2. 6 0 obj The Calculus of Variations is concerned with solving Extremal Problems for a Func-tional. x��YMo�F��W�H!�f�? Minimization problems that can be analyzed by the calculus of variationsserve tochar- endobj %PDF-1.5 Time-dependent equations in which u0 = du=dt. 0W!`�{�Ep��-�L��@��A=��Гبν�g�G�{ܜ��6!�*����Osx���� �I�����A����%� ����&�.b]3‟Yz!�Ui�B�L�6?4z@N�p����f��γ}Ov?n�*��@[P��s�#ݛj������ĵ��a��gG�bP�Ao����ܬ�+g}\04�. x��V�n�6��;�(- ���*b%^�h�nkt����1�ĭ�d���̐�eY�=H�~�?$%�~��g��ϗBU��]����D %�R@y�A g�ح�����t��7�-�}�N>M'���cV�۬�9 y(c):y(x,c)x1≤ x ≤ x2(c)− δ ≤ c ≤ δ(20b) (where only the right end value ofxvaries withcsince the left end value is fixed and) which safisfies the Euler equation and. G�$�B�7� The condition is equivalent to the zero gradient condition for the discrete time case. For a quadratic P( u) = 1 2 TKu − Tf, there is no … 1 0 obj << carries ordinary calculus into the calculus of variations. z3�����I2����#��ۆ��ࠇ���(���� s85��H����`a�i��t����LVZ�d:����X�$�Y�֎��GC���e�`�aN�4L��%/� �s�C�fg�[�w8��u����=^�j�e��އ5�;�&%>652�|g�. That is to say Maximum and Minimum problems for functions whose domain con-tains functions, Y(x) (or Y(x1;¢¢¢x2), or n-tuples of functions). /Length 1278 * ªº¬¼x x dx³cos. ��� ���U�#h�O�q���͟��M�&�@e���ö{: ��_d�'��.&�:Y)��=��>��h�n�e(�A�}����$���j�n��3fX�Cn����Y����앃i"�h�"A�N�=��+�Z249Gqi_��F�j��c����������|�9�|B'˞� 蘛�{��a�����W�y�q֩���� ��a���b�l/2q���]���چ��a+_]�(��Ŭ��6#�gi&RUȧ�NHo����C"W�bG1�����!��>jR�%S�=��)�~��� ��(���35 N ��=m帞p �X!zF���_r�Al3����1X]��i�FL����E�Pg}2��Y���0ٿRNg���p.��Z����Kf��I�N�^,�x� j��0���SD�b��8brȘ�M�L�T�����=O�ý��sj��2r_X�ibՇc��t���ɾ��� %PDF-1.5 <> stream endobj /Length 3725 /Filter /FlateDecode ����z>AE������|�D�t ���A��}�eos�ל�P����sy�t'Q�B]J���jP�����_�n�*F�@�$ The goal is to find a y(x) that minimizes Г, or maximizes it. endobj {��jU���q�t�@��p�Y!��F�I�jĤ�� C��-��ۊ*,r/.��Ƈ�H�g����. /MediaBox [0 0 612 792] 3 0 obj ($isj�� ˲+@�Jl����%)RZQ4�&-z�V�pvv�ٙ!ͮ3��:��'�_�����x����/��#N)��_eLb�ʹ2���_f�rF�h�W����b1[�ч�׍�/m��\{ 43K�u���o���X����B�/6��:�o�i3)��*~{O�\]�H�n��l9Y���_���b�(F��_���6��W;YxKqȖ��Q"����P&��Vr4��a��b�H]I]x�l,���0#�/��|�Rߩ1�y�I����ׂ���C�j~�{OMhq� ޣD�%Ёxr��'��R.�H}�C�_��}���2���a62�.��}��%~{�Q"�;ɥ?��E��� After that, going from two to three was just more algebra and more complicated pictures. ���t�k7�4��K���OIp *��������Ѥ�� Two-dimensional problems P (u) = F (u, ux, uy) dx dy 4. Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. /Filter /FlateDecode 7.2. A branch of mathematics that is a sort of generalization of calculus. endobj Two-dimensional problems P(u) = RR F(u;ux;uy)dxdy 4. /Parent 15 0 R stream >> endobj y(x1,c)=y1. �C�lޘ��c�5�T+h��PT�6SD�>�)�;�L%r��s}�q4I��'CS�4:��x6yl��������0y�ӷo�}�M�G1�?��b��V� �&���~ȁSv�����b��|��"F~���/�|N���4[^�Y� Calculus of variations seeks to find the path, curve, surface, etc., for which a given function has a stationary value (which, in physical problems, is usually a minimum or maximum). Calculus of Variations solvedproblems Pavel Pyrih June 4, 2012 ( public domain ) Acknowledgement.The following problems were solved using my own procedure in a program Maple V, release 5. The fundamental lemma of calculus of variations states that if Z b a m(x)g(x)dx= 0 (14) for all gwith continuous second partial derivatives, then m(x) = 0;for x2(a;b) (15) Variational Methods The basic problem of the calculus of variations is to determine the function that extremizes a functional. >> <> I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. stream <> I would have included the Euler-Lagrange proof but it would have likely made the video over 15 minutes, which is a little too long for us.Questions/requests? <> 7 0 obj stream %�쏢 %���� stream Note: a functional is a function of functions, or a map which converts a function to a real number, kind of like how the time functional T converts a particular function/particle path to a real value of time. 9 0 obj %PDF-1.5 In an inhomogeneous planar optical medium, the speed of light, c(x,y), varies from point to point, depending on the optical properties. ?�~����WW?��3��x��=@s��Ȭ*U 'XkM)��p2���f�9ŽF���Q���3j,o.F��$Q�_��.,&�L7��ֆ��I�g Fd���A`��V��f}���-�o f{��B�J>�)z�59t%��ɇ�]D�.h�]8�����QVa��� �;��Q�y�Ӫ�X�B����l�=k^8IU��B�Q�+lG�i �We��~�����j�NJ/�w� +�$VR �h�O�u+x]J E�$��x��������c�a�e�����/u��c$��D2�:IT���\��gO�2�z��������9y�E`V@(p��L}���Ⱦ�@i=�Ę�~ؾ]uk����*�%�uds77�n2�`�De9��0\P�l��C�l���y3�r��#B�*A�2����P��FY�l�]���S��}e�כe�� ��Ķ��3z��� �H�$�CD8�G$��yA�X�8������)Ў��b�������^lCu��Ͳ�9`���9��vD�7}���ݡ]��-Z�Y�&2,��L�*�$��vhW�D ��Ij��E���fE�ó�� ��!�R��7� �����?�7�~��!a���G��J� ��� ���o�݀�Cf 43ʯ�] ��q3�ߧ���-�rLj�����L\�nF� ��$��6�l�����n����]�v�$h b��6Q1�&�#��s���K!��C��ߡ. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 10 0 R/Group<>/Tabs/S/StructParents 1>> * ªº¬¼f x F x y x y x dx. Speed equals the time derivative of distance traveled, namely, the arc length of the curve y = u(x) traced by the light ray. stream >> <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 1 22 1 0 0. /Font << /F16 4 0 R /F8 5 0 R /F18 6 0 R /F19 7 0 R /F11 8 0 R /F13 9 0 R /F1 10 0 R /F10 11 0 R /F7 12 0 R /F14 13 0 R /F31 14 0 R >> XE����q�'/�Ԗ~�|v���P�.3S�S���-cq0Z�X�Rce�e�"·el�(eX;ӷhE������ޜf"`& ]?\�m�FO�I��%K�i�=P �{�J�F��~M�*�� A �O��8�Eo$�P�ؚ�=���(P�wox�����y��D^.q��Y�4r!��V�H�#no�&�/J&��, �c_���4��)�j �"ai�����m�!�n�0?�L��k�ƈ�2-��V%��Ӂ���LN���EE���g��i�Ѣ�,�~�ܢ��� <> Javier R. Movellan Copyright c 2003 Javier R. Movellan. >> ]^�I15a��A��-�Ǘbx|�MHb�#g�d �� �����XX�NF� Constraints, not necessarily linear, with their Lagrange multipliers 3. xڭZ�o��~Ţ_,�^EԋR�C�K.�I��N�6 PyE�2�cO����ϼ��-(�/+�����oF����g_'j�"?�������(��,ڥy�q��-w�x_�a���zE����3��u��^��b�m�_�v�-0�w*��$ˑ�^E���d�#_���ݞ0�������̙[�][c+�J��������p�V��KqfHqWI�����ٱW����Bf���+�>��,��� We do it in several steps: 1. There may be more to it, but that is the main point. /Type /Page Calculus of Variations. carries ordinary calculus into the calculus of variations. In this video, I introduce the subject of Variational Calculus/Calculus of Variations. %PDF-1.3 endobj 3 0 obj << Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. Now the step will be from a nite number of variables to an in nite number. endobj Mathematically, this involves finding stationary values of integrals of the form I=int_b^af(y,y^.,x)dx. 10 0 obj To minimize P is to solve P = 0. x��WKo�F��W�q��}?ZhZ�EHc!�Z�l"�������>H�2��q���7�̮(�C�yA_x�����bqI�5-��9M3HKG�Uh�A+�G^���k�������0�M��E�p�o7ˏ`�"���/>�ݞA�jP�-a�Gg���Â0�ą�Y�z"B÷`� "տ�n���X�� �(����U���ԷU�kϼ���E ��ڃ�1Z �8f#�������?=5���I���;���QQ��O�9e�%}J?3�* �۪\G#��{竞����]�e��L�2&�Z�uC�K�z�� ~����Xޢ�}V�b����^�^ރWo몊�^'�s翘S�ۣ�)�"��u�66�k˨��������~���Ŧk5 ��Z*�S�#Vy�|���U�ŗT�����M��,2)%~WUI�\�>����Ģ��,��i��&m�|�Oqs�wձ �A�[�$Rs�51������ a>zp��}WM\�;��z�B���Q�@��1�T�n�]�h�}i e�����K��_w�C��n���I����������!�5����+ ����E�K�Ht���"THP�0B�*}��)����P��H�2�7ES>,��P�H�6~��}�Ov#p�l|U��a��[��~c8 ���;�����[�p�}���o��v�n��b��L���v�v�܋�_Q�Os\ᒏl0��o��qJ�0e����~NBtDM4��8�0ރ�m*q�=GM�#�K�(� h%��Jo�z�j�� �t(Ϗ�,�� \:�2�'�8�ǣ�N���p6����ӡ9Ҥ�q��������#B�3R��X��L����D4y]B[�g��}����3~$R�����.4T�SB�F��O�f��m�����E��p"�_^|O�C\3�8C�; �C���ꆢ |�3�g�c�����e��/�NW� V`KB� F�c��&��ZD�A��Ӱ~N���L�똞kv�&��n�� ����j( >��6��6?Դ��g�R[s��L�� Ƴ$ؿ:�҄��MbF��C��yxv\՛����mbk���2�^�pUQı�Ԃ�*~��ܕ���i��s��`��Ft�0ۗ��9F�a\�: The range of the functional will be the real numbers, R Examples: I. Find the curve between two given points in the plane that yields a surface of revolution %�쏢 /Filter /FlateDecode Calculus of Variations. endobj �E�``.-�ހ�����qy_���S1�0ݱ�͑��S�'��M�,��+z~�}$�A�ni�����QG��ES�tZ�2������������3�O[֛��e��Q�H�#Qxd�{����V�]�a%�/�G (���@ōd�k!︃���`4̵�*Z��R P ���E�pIx����g9;�q���D�1+� @{%y%�_���{����z����zv��{2��*�=~q���A���PҚ�_�� �T���9{x���g��]��Ca��嵖�u^��]�#�`�����8bT^��+�TF�pz��T�Ҝ������ku��җ����o��R 6 0 obj ����(�t��>����W�����+{x�xo��V�^�C;i��y�O�̎�'O��@Za������%>40�;��u��#Q����� ��{'���~� ���/`e���S�9�H)�h�J[ �Ս����_=x���k�a����C[�t�.��ƕg���6��:�>�� �Rw:ࡆ5��tm)UpH��5Y*cz�i�зu�(����{����_�����@s��a��3��w A���y;O�ħ�0� mm����=Y[����NJ��]Y$��mq��c��0��*+RAȺ����>SB�N��($�d)�{�A{��ݵr����.Ձ`ƙ ��V� �[�I u��C��"�Czشq�KG�K|\(�>�����#4�C�l��4�S;��F r3n)ʞ�"Z�!�2�\`볧�כ���ڳ��ً��e�u����'kۙ�_�O޵޵g�ۯio?i�>iϾoϦ_��ў��?G�92�c����ekq0����P?������q�"�I�LI��cUF�vFЬB�������wi|�'�]�&�"I��A�ru0r��o ��z� 4e �A�~�����_���G=�� *T

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