query: Spivak's Calculus on Manifolds - Problem 1-30.
positive_passages.docid: 85465
positive_passages.text: Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.
negative_passages.docid: 52470
negative_passages.text: Spivak's Calculus - Exercise 4.a of 2nd chapter
negative_passages.docid: 101418
negative_passages.text: Can we define flat connection on any given smooth manifold?
negative_passages.docid: 132063
negative_passages.text: locally on $S$ one of the coordinates is a $C^∞$ function of the other coordinate. Show that $S$ is a submanifold of $R^2$
negative_passages.docid: 137093
negative_passages.text: What is the codimension of matrices of rank $r$ as a manifold?
negative_passages.docid: 144548
negative_passages.text: topological structure on smooth manifolds
negative_passages.docid: 103397
negative_passages.text: Homeomorphisms between infinite-dimensional Banach spaces and their spheres
negative_passages.docid: 123478
negative_passages.text: Generalized Poincaré Inequality on H1 proof.
negative_passages.docid: 124861
negative_passages.text: The extension of smooth function
negative_passages.docid: 78624
negative_passages.text: CW complexes and manifolds
negative_passages.docid: 120130
negative_passages.text: Baker-Hausdorff Lemma from Sakurai's book
negative_passages.docid: 154009
negative_passages.text: Closed, simply connected manifolds which are not spheres
negative_passages.docid: 84419
negative_passages.text: Product of manifolds & orientability
negative_passages.docid: 64463
negative_passages.text: Proof that angle-preserving map is conformal
negative_passages.docid: 3201
negative_passages.text: Landau & Lifshitz's Approach (contour method) on the WKB connection formulas
negative_passages.docid: 152725
negative_passages.text: Why does $S^n$ satisfy the local $n$-slice condition? (From Lee's Smooth Manifolds)
negative_passages.docid: 127631
negative_passages.text: On a conformal mapping