{ localUrl: '../page/math_3_examples.html', arbitalUrl: 'https://arbital.com/p/math_3_examples', rawJsonUrl: '../raw/56b.json', likeableId: '2986', likeableType: 'page', myLikeValue: '0', likeCount: '4', dislikeCount: '0', likeScore: '4', individualLikes: [ 'AlexeiAndreev', 'JaimeSevillaMolina', 'MarkChimes', 'EricRogstad' ], pageId: 'math_3_examples', edit: '7', editSummary: '', prevEdit: '6', currentEdit: '7', wasPublished: 'true', type: 'wiki', title: 'Math 3 example statements', clickbait: 'If you can read these formulas, you're in Math 3!', textLength: '2012', alias: 'math_3_examples', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'JoeZeng', editCreatedAt: '2016-07-10 12:29:15', pageCreatorId: 'JoeZeng', pageCreatedAt: '2016-07-07 01:09:50', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '4', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '381', text: 'If you're at a Math 3 level, you'll probably be familiar with at least some of these sentences and formulas, or you would be able to understand what they meant on a surface level if you were to look them up. Note that you don't necessarily have to understand the *proofs* of these statements (that's what we're here for, to teach you what they mean), but your eyes shouldn't gloss over them either.\n\n> In a [3gd group] $G$, the [-4bj] of an element $g$ is the set of elements that can be written as $hgh^{-1}$ for all $h \\in G$.\n\n> The [ rank-nullity theorem] states that for any [-linear_mapping] $f: V \\to W$, the [-dimension] of the [-image] of $f$ plus the dimension of the [-kernel] of $f$ is equal to the dimension of $V$.\n\n> A [ Baire space] is a space that satisfies [ Baire's Theorem] on [complete_metric_space complete metric spaces]: For a [-topological_space] $X$, if ${F_1, F_2, F_3, \\ldots}$ is a [countable_set countable] collection of open sets that are [dense_set dense] in $X$, then $\\bigcap_{n=1}^\\infty F_n$ is also dense in $X$.\n\n> The [riemann_hypothesis] asserts that every non-trivial zero of the [riemann_zeta_function] $\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{s^n}$ when $s$ is a complex number has a real part equal to $\\frac12$.\n\n> $\\newcommand{\\pd}[2]{\\frac{\\partial #1}{\\partial #2}}$ The [jacobian_matrix] of a [vector_valued_function vector-valued function] $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ is the matrix of [-partial_derivatives] $\\left[ \\begin{matrix} \\pd{y_1}{x_1} & \\pd{y_1}{x_2} & \\cdots & \\pd{y_1}{x_m} \\\\ \\pd{y_2}{x_1} & \\pd{y_2}{x_2} & \\cdots & \\pd{y_2}{x_m} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\pd{y_n}{x_1} & \\pd{y_n}{x_2} & \\cdots & \\pd{y_n}{x_m} \\end{matrix} \\right]$ between each component of the argument vector $x = (x_1, x_2, \\ldots, x_m)$ and each component of the result vector $y = f(x) = (y_1, y_2, \\ldots, y_n)$. It is notated as $\\displaystyle \\frac{d\\mathbf{y}}{d\\mathbf{x}}$ or $\\displaystyle \\frac{d(y_1, y_2, \\ldots, y_n)}{d(x_1, x_2, \\ldots, x_m)}$.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: [ '0', '0', '0', '0', '0', '0', '0', '0', '0', '0' ], muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'JoeZeng', 'DylanHendrickson' ], childIds: [], parentIds: [ 'math3' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: 'math3', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16364', pageId: 'math_3_examples', userId: 'JoeZeng', edit: '7', type: 'newEdit', createdAt: '2016-07-10 12:29:15', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16285', pageId: 'math_3_examples', userId: 'JoeZeng', edit: '6', type: 'newEdit', createdAt: '2016-07-09 01:29:51', auxPageId: '', oldSettingsValue: '', newSettingsValue: 'Tried to salvage the vector space statement.' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16275', pageId: 'math_3_examples', userId: 'DylanHendrickson', edit: '5', type: 'newEdit', createdAt: '2016-07-08 23:14:15', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16274', pageId: 'math_3_examples', userId: 'DylanHendrickson', edit: '4', type: 'newEdit', createdAt: '2016-07-08 23:13:29', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16003', pageId: 'math_3_examples', userId: 'JoeZeng', edit: '3', type: 'newEdit', createdAt: '2016-07-07 17:13:36', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15881', pageId: 'math_3_examples', userId: 'JoeZeng', edit: '2', type: 'newEdit', createdAt: '2016-07-07 01:12:24', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15879', pageId: 'math_3_examples', userId: 'JoeZeng', edit: '0', type: 'newParent', createdAt: '2016-07-07 01:10:19', auxPageId: 'math3', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15877', pageId: 'math_3_examples', userId: 'JoeZeng', edit: '1', type: 'newEdit', createdAt: '2016-07-07 01:09:50', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }