{"id":1513,"date":"2017-11-20T12:54:30","date_gmt":"2017-11-20T17:54:30","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1513"},"modified":"2017-11-20T12:54:30","modified_gmt":"2017-11-20T17:54:30","slug":"daily-definitions-from-class-on-111617","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/20\/daily-definitions-from-class-on-111617\/","title":{"rendered":"Daily Definitions (From class on 11\/16\/17)"},"content":{"rendered":"<p>In this class, the topic of Riemann Integration was introduced by Dr. K. This post will cover the definitions and lemmas that were covered during class. Note that the definitions build on another because we slowly built up to the idea of being Riemann Integrable.<\/p>\n<p><span style=\"color: #000000\">1. A function f is <strong>bounded on [a,b] <\/strong>if <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=sup%5C%7Bf%28x%29%3A+x+%5Cin+%5Ba%2Cb%5D%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"sup&#92;{f(x): x &#92;in [a,b]&#92;}\" class=\"latex\" \/> is finite and <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=inf%5C%7Bf%28x%29%3Ax+%5Cin+%5Ba%2Cb%5D%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"inf&#92;{f(x):x &#92;in [a,b]&#92;}\" class=\"latex\" \/> is finite.<\/span><\/p>\n<p><span style=\"color: #000000\">2. <strong>Partition P of [a,b]<\/strong> is a set of numbers <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7Bx_0%2C+x_1%2C+%5Cdots+%2C+x_n%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{x_0, x_1, &#92;dots , x_n&#92;}\" class=\"latex\" \/> with <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=x_o+%3D+a%2C%5C+x_%7Bk-1%7D+%3Db%2C+x_%7Bk-1%7D+%3Cx_k%5C+%5Cforall+%5C+k+%5Cin+1%2C+%5Cdots%2C+n+&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"x_o = a,&#92; x_{k-1} =b, x_{k-1} &lt;x_k&#92; &#92;forall &#92; k &#92;in 1, &#92;dots, n \" class=\"latex\" \/> <\/span><\/p>\n<ul>\n<li><span style=\"color: #000000\"><em>Notation:<\/em> The set of all possible partitions of [a,b] is denoted <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cmathscr+%7BP%7D_%7B%5Ba%2Cb%5D%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;mathscr {P}_{[a,b]}\" class=\"latex\" \/><\/span><\/li>\n<\/ul>\n<p><span style=\"color: #000000\">3. A partition Q is a <strong>refinement<\/strong> <\/span>of a given partition P if <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=Q+%5Csubseteq+P.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"Q &#92;subseteq P.\" class=\"latex\" \/><\/p>\n<ul>\n<li>\u00a0<em><span style=\"color: #008000\"><span style=\"color: #0000ff\">Lemma:<\/span>\u00a0<\/span><\/em><span style=\"color: #008000\"><span style=\"color: #000000\">Under certain conditions, if Q is a refinement of P, then\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%7Cq-%5Cint_%7Ba%7D%5E%7Bb%7D+f%7C+%5Cleq+%7Cp-%5Cint_%7Ba%7D%5E%7Bb%7D+f%7C.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"|q-&#92;int_{a}^{b} f| &#92;leq |p-&#92;int_{a}^{b} f|.\" class=\"latex\" \/> <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=q+%5Cin+Q%2C+p+%5Cin+P&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"q &#92;in Q, p &#92;in P\" class=\"latex\" \/><\/span><\/span><\/li>\n<\/ul>\n<p>4.\u00a0Given <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%2C+%5Ba%2Cb%5D%2C+P%2C+m_k%2C+%5C+and%5C+M_k%2C&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f, [a,b], P, m_k, &#92; and&#92; M_k,\" class=\"latex\" \/> we can construct the lower and upper sums for <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f\" class=\"latex\" \/> over <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Ba%2Cb%5D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"[a,b]\" class=\"latex\" \/> using P. Note: <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=m_k+%3A%3D+inf%5C%7Bf%28x%29%3A+x%5Cin+%5Bx_%7Bk-1%7D%2Cx_k%5D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"m_k := inf&#92;{f(x): x&#92;in [x_{k-1},x_k]\" class=\"latex\" \/> and <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=M_k+%3A%3D+sup%5C%7Bf%28x%29%3A+x%5Cin+%5Bx_%7Bk-1%7D%2Cx_k%5D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"M_k := sup&#92;{f(x): x&#92;in [x_{k-1},x_k]\" class=\"latex\" \/><\/p>\n<ul>\n<li><strong>Lower Sum<\/strong> for f over [a,b] using P:\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=L%28f%2C%5Ba%2Cb%5D%2CP%29%3D+%5Csum_%7Bk%3D1%7D%5E%7Bn%7D+m_k+%28x_k-x_%7Bk-1%7D%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"L(f,[a,b],P)= &#92;sum_{k=1}^{n} m_k (x_k-x_{k-1})\" class=\"latex\" \/><\/li>\n<li><strong>Upper Sum<\/strong>\u00a0for f over [a,b] using P:\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=U%28f%2C%5Ba%2Cb%5D%2CP%29%3D+%5Csum_%7Bk%3D1%7D%5E%7Bn%7D+M_k+%28x_k-x_%7Bk-1%7D%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"U(f,[a,b],P)= &#92;sum_{k=1}^{n} M_k (x_k-x_{k-1})\" class=\"latex\" \/>\n<ul>\n<li><span style=\"color: #0000ff\"><em>Lemma:<\/em><\/span> For a lower sum built from P, the refinement Q of P, then the lower sum of Q satisfies L(f,[a,b], Q) \u2265 L(f,[a,b],P). Similarly, for upper bounds, U(f, [a,b], Q) \u2264 U(f, [a,b], P).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>5.<\/p>\n<ul>\n<li><strong>U(f)<\/strong>: <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=U%28f%2C%5Ba%2Cb%5D%29+%3D+inf+%5C%7BU+%28f%2C+%5Ba%2Cb%5D%2C+P%29%3A+P+%5Cin+%5Cmathscr+%7BP%7D_%7B%5Ba%2Cb%5D%7D+%5C%7D+%3A%3D+f_%7Bupper%5C+over%5C+%5Ba%2Cb%5D%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"U(f,[a,b]) = inf &#92;{U (f, [a,b], P): P &#92;in &#92;mathscr {P}_{[a,b]} &#92;} := f_{upper&#92; over&#92; [a,b]}\" class=\"latex\" \/><\/li>\n<li><strong>L(f)<\/strong>: <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=L%28f%2C%5Ba%2Cb%5D%29+%3D+sup+%5C%7BL+%28f%2C+%5Ba%2Cb%5D%2C+P%29%3A+P+%5Cin+%5Cmathscr+%7BP%7D_%7B%5Ba%2Cb%5D%7D+%5C%7D+%3A%3D+f_%7Blower%5C+over%5C+%5Ba%2Cb%5D%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"L(f,[a,b]) = sup &#92;{L (f, [a,b], P): P &#92;in &#92;mathscr {P}_{[a,b]} &#92;} := f_{lower&#92; over&#92; [a,b]}\" class=\"latex\" \/><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>6. A function f defined on [a,b] is Riemann integrable if <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=L%28f%29+%3D+U%28f%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"L(f) = U(f)\" class=\"latex\" \/>.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this class, the topic of Riemann Integration was introduced by Dr. K. This post will cover the definitions and lemmas that were covered during class. Note that the definitions build on another because we slowly built up to the idea of being Riemann Integrable. 1. A function f is bounded on [a,b] if is [&hellip;]<\/p>\n","protected":false},"author":2206,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58823],"tags":[],"class_list":["post-1513","post","type-post","status-publish","format-standard","hentry","category-definitions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-op","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1513","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/2206"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1513"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1513\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1513"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1513"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1513"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}