Ebook All the mathematics you missed: but need to know for graduate school - Part 2
➤ Gửi thông báo lỗi ⚠️ Báo cáo tài liệu vi phạmNội dung chi tiết: Ebook All the mathematics you missed: but need to know for graduate school - Part 2
Ebook All the mathematics you missed: but need to know for graduate school - Part 2
Chapter 7Curvature for Curves andSurfacesBasic Objects:Curves and surfaces in spaceBasic Goal:Calculating curvaturesMost of high school mathematics is Ebook All the mathematics you missed: but need to know for graduate school - Part 2 s concerned with straight lines and planes. There is of course far more to geometry than these flat objects. Classically differential geometry is concerned with how curves and surfaces bend and twist in space. The word “curvature” is used to denote the various measures of twisting that have been dis Ebook All the mathematics you missed: but need to know for graduate school - Part 2 covered.Unfortunately, the calculations and formulas to compute the different types of curvature are quite involved and messy, but whatever curvatureEbook All the mathematics you missed: but need to know for graduate school - Part 2
is, it should be the case that the curvature of a straight line and of a plane must be zero, that the curvature of a circle (and of a sphere) of radiuChapter 7Curvature for Curves andSurfacesBasic Objects:Curves and surfaces in spaceBasic Goal:Calculating curvaturesMost of high school mathematics is Ebook All the mathematics you missed: but need to know for graduate school - Part 2 ius circle (or sphere) (which captures the idea that it is easier to balance on the surface of the earth than on a bowling ball).The first introduction to curvature-type ideas is usually in calculus. While the first derivative gives us tangent line (and thus linear) information, it is the second der Ebook All the mathematics you missed: but need to know for graduate school - Part 2 ivative that measures concavity, a curvature-type measurement. Thus we should expect to see second derivatives in curvature calculations.7.1 Plane CurEbook All the mathematics you missed: but need to know for graduate school - Part 2
ves146 CHAPTER 7. CURVATURE FOR CURVES■r(i) = (x(t),7/(i))r : R -> R2.and thus as a mapt-axisThe variable t is called the parameter (and is frequentlyChapter 7Curvature for Curves andSurfacesBasic Objects:Curves and surfaces in spaceBasic Goal:Calculating curvaturesMost of high school mathematics is Ebook All the mathematics you missed: but need to know for graduate school - Part 2 )both describe a unit circle. Any calculation of curvature should be independent of the choice of parametrization. There are a couple of reasonable ways to do this, all of which can be shown to be equivalent. We will take the approach of always fixing a canonical parametrization (the arc length para Ebook All the mathematics you missed: but need to know for graduate school - Part 2 metrization). This is the parametrization r : [a, b] -> R such that the arc length of the curve is just b - a. Since the arc length iswe nccd ự(df)2 +Ebook All the mathematics you missed: but need to know for graduate school - Part 2
(df)2 = !•Thus for the arc length parametrization,the length of the tangent vector must always be one:|T(s)| =drdsdx dy\ d? ds/Note that each point oChapter 7Curvature for Curves andSurfacesBasic Objects:Curves and surfaces in spaceBasic Goal:Calculating curvaturesMost of high school mathematics is Ebook All the mathematics you missed: but need to know for graduate school - Part 2 ying to define curvature as a measure of the change in the direction of the tangent vectors. To measure a rate of change we need to use a derivative. This leads to:Definition 7.1.1 For a plane curve parametrized by arc lengthr(s) = (z(s),2/(s)),define the principal curvature K at a point on the curv Ebook All the mathematics you missed: but need to know for graduate school - Part 2 e to be the length of the derivative of the tangent vector with respect to the parameter s, i.e.j«= dT(5) .dsConsider the straight line r(s) = (asib.cEbook All the mathematics you missed: but need to know for graduate school - Part 2
si d), where a,b,c and d are constants. The tangent vector is:drChapter 7Curvature for Curves andSurfacesBasic Objects:Curves and surfaces in spaceBasic Goal:Calculating curvaturesMost of high school mathematics isChapter 7Curvature for Curves andSurfacesBasic Objects:Curves and surfaces in spaceBasic Goal:Calculating curvaturesMost of high school mathematics isGọi ngay
Chat zalo
Facebook