Uniform hyperbolicity of the curve graphs
... so α cuts Σ into two discs, H0 , H1 . We have |Π ∩ Hi | ≥ 2, and we can assume that |Π ∩ H0 | ≥ 3. Note also, if αi′ ∪ δi is non-trivial, then length(αi′ ∪ δi ) ≥ length(α) and so length(αi ) ≤ length(δi ) < λ. Now H0 must contain at least two bridges from our collection. We can assume these are δ1 ...
... so α cuts Σ into two discs, H0 , H1 . We have |Π ∩ Hi | ≥ 2, and we can assume that |Π ∩ H0 | ≥ 3. Note also, if αi′ ∪ δi is non-trivial, then length(αi′ ∪ δi ) ≥ length(α) and so length(αi ) ≤ length(δi ) < λ. Now H0 must contain at least two bridges from our collection. We can assume these are δ1 ...
Get Notes - Mindset Learn
... turning point and then to substitute it into the equation for the corresponding y value. The other alternative is to complete the square which can be very difficult. ...
... turning point and then to substitute it into the equation for the corresponding y value. The other alternative is to complete the square which can be very difficult. ...
Practice Explanations: Solutions 1. Suppose y1 and y2 are both
... In this case, it is obvious, and so we can always find c1 and c2 so that y will satisfy the initial conditions. This is more thorough than you would have to be, but: In general, the columns are (y1 (0), y10 (0)) and (y2 (0), y20 (0)). If these were multiples of each other, then there would be some n ...
... In this case, it is obvious, and so we can always find c1 and c2 so that y will satisfy the initial conditions. This is more thorough than you would have to be, but: In general, the columns are (y1 (0), y10 (0)) and (y2 (0), y20 (0)). If these were multiples of each other, then there would be some n ...
Chapter Projects
... (a) The aphelion of Neptune is 4532.2 * 106 km and its perihelion is 4458.0 * 106 km. Find the equation for the orbit of Neptune around the Sun. (b) The aphelion of Pluto is 7381.2 * 106 km and its perihelion is 4445.8 * 106 km. Find the equation for the orbit of Pluto around the Sun. (c) Graph the ...
... (a) The aphelion of Neptune is 4532.2 * 106 km and its perihelion is 4458.0 * 106 km. Find the equation for the orbit of Neptune around the Sun. (b) The aphelion of Pluto is 7381.2 * 106 km and its perihelion is 4445.8 * 106 km. Find the equation for the orbit of Pluto around the Sun. (c) Graph the ...
Why MR is below the Demand Curve
... additional revenue added by selling an additional unit of output: Marginal Revenue = (Change in total revenue) divided by (Change in sales) According to the picture, people will not buy more than 100 units at a price of $10.00. To sell more, price must drop. Suppose that to sell the 101st unit, the ...
... additional revenue added by selling an additional unit of output: Marginal Revenue = (Change in total revenue) divided by (Change in sales) According to the picture, people will not buy more than 100 units at a price of $10.00. To sell more, price must drop. Suppose that to sell the 101st unit, the ...
Conics - Parabolas
... a plane, the plane will pass through the base of the cone, but will only pass through one of the cones. Any of these conics can be graphed on a coordinate plane. The graph of any conic on an x-y coordinate plane can be described by an equation in the form: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 This equa ...
... a plane, the plane will pass through the base of the cone, but will only pass through one of the cones. Any of these conics can be graphed on a coordinate plane. The graph of any conic on an x-y coordinate plane can be described by an equation in the form: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 This equa ...
Gauss` Theorem Egregium, Gauss-Bonnet etc. We know that for a
... non-trivial kernel of L. If this kernel is the whole tangent space, at each point of S, the second fundamental form is zero and it is easy to show that S is contained in a plane. Suppose on the other hand that this kernel has dimension 1 (that is, the second fundamental from is not zero). We can cho ...
... non-trivial kernel of L. If this kernel is the whole tangent space, at each point of S, the second fundamental form is zero and it is easy to show that S is contained in a plane. Suppose on the other hand that this kernel has dimension 1 (that is, the second fundamental from is not zero). We can cho ...
Math 106: Course Summary
... by kdB/dsk. That is, we can measure how fast the osculating planes are spinning around by measuring the rate of change of B. The Frenet-Serret Equations Continuing with the discussion for space curves, the vectors {T, N, B} form an orthonormal basis at each point along γ (as long as everything is de ...
... by kdB/dsk. That is, we can measure how fast the osculating planes are spinning around by measuring the rate of change of B. The Frenet-Serret Equations Continuing with the discussion for space curves, the vectors {T, N, B} form an orthonormal basis at each point along γ (as long as everything is de ...
File
... graph on either side of the axis of symmetry look like mirror images of each other. The x-intercepts are the points or the point at which the parabola intersects the x-axis. A parabola can have either 2,1 or zero real xintercepts. The y-intercept is the point at which the parabola intercepts the yax ...
... graph on either side of the axis of symmetry look like mirror images of each other. The x-intercepts are the points or the point at which the parabola intersects the x-axis. A parabola can have either 2,1 or zero real xintercepts. The y-intercept is the point at which the parabola intercepts the yax ...
Problem set done in class
... glides across the snow-covered, horizontal surface. [Show all work, including the equation and substitution with units.] [2] ...
... glides across the snow-covered, horizontal surface. [Show all work, including the equation and substitution with units.] [2] ...
Coordinates, points and lines
... We can tell whether or not a point is on a line (or curve) by substituting the coordinates into the equation and seeing if the equation is satisfied by the coordinates of the point. The equation of a line or curve is a rule for determining whether or not the point with coordinates x, y lies on t ...
... We can tell whether or not a point is on a line (or curve) by substituting the coordinates into the equation and seeing if the equation is satisfied by the coordinates of the point. The equation of a line or curve is a rule for determining whether or not the point with coordinates x, y lies on t ...
lecture 2
... representations which are widely used in CAD and computer graphics will be discussed later. The representations of curves lead naturally to representations of surfaces. Points and vectors ...
... representations which are widely used in CAD and computer graphics will be discussed later. The representations of curves lead naturally to representations of surfaces. Points and vectors ...
Section 4.3 Line Integrals - The Calculus of Functions of Several
... This is the result we should expect: as long as the curve is traversed only once, the work done by a force when an object moves along the curve should depend only on the curve and not on any particular parametrization of the curve. We need to verify the previous statement in general before we can s ...
... This is the result we should expect: as long as the curve is traversed only once, the work done by a force when an object moves along the curve should depend only on the curve and not on any particular parametrization of the curve. We need to verify the previous statement in general before we can s ...
Problem. For the ODE dy dx = y - 4x x
... through basic algebraic manipulation. Note that we are implicitly making the assumption that x 6= 0 in our analysis. However, since we are simply looking at the structure of the ODE for clues as to how to solve it, this is okay. We may have to separately study the cases for initial values y(0) = y0 ...
... through basic algebraic manipulation. Note that we are implicitly making the assumption that x 6= 0 in our analysis. However, since we are simply looking at the structure of the ODE for clues as to how to solve it, this is okay. We may have to separately study the cases for initial values y(0) = y0 ...
Formula Sheet, PHYS 101, Final Exam Includes the important
... 4) Equation 9.7, Energy-mass conversion equation: E = mc2 m is the mass converted to energy (in kg), c is the speed of light, E is the amount of energy released (in Joules). 5) Equation 9.13, Proton-Proton chain, first step: 1H + 1H → 2H + e+ + ν H is Hydrogen, e+ is a positron, ν is a neutrino. 6) ...
... 4) Equation 9.7, Energy-mass conversion equation: E = mc2 m is the mass converted to energy (in kg), c is the speed of light, E is the amount of energy released (in Joules). 5) Equation 9.13, Proton-Proton chain, first step: 1H + 1H → 2H + e+ + ν H is Hydrogen, e+ is a positron, ν is a neutrino. 6) ...
Day 2- Continued Example 4: Find all possible value(s) of b such
... Example 4: Find all possible value(s) of b such that h(x) = !6x 2 ! 24x + b is tangent to the function g(x) = 3x 4 . ...
... Example 4: Find all possible value(s) of b such that h(x) = !6x 2 ! 24x + b is tangent to the function g(x) = 3x 4 . ...
Lesson 4 - Solving Multiplication Equations
... Lesson 9-4 Example 1 Solve a Multiplication Equation Solve 6x = 18. Check your solution. ...
... Lesson 9-4 Example 1 Solve a Multiplication Equation Solve 6x = 18. Check your solution. ...
When solving a fixed-constant linear ordinary differential equation
... When solving a fixed-constant linear ordinary differential equation where the characteristic equation has repeated roots, why do we get the next independent solution in the form of x n e mx ? Show this through an example. Let’s suppose we want to solve the ordinary differential equation d2y dy + 6 + ...
... When solving a fixed-constant linear ordinary differential equation where the characteristic equation has repeated roots, why do we get the next independent solution in the form of x n e mx ? Show this through an example. Let’s suppose we want to solve the ordinary differential equation d2y dy + 6 + ...
Catenary
In physics and geometry, a catenary[p] is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The curve has a U-like shape, superficially similar in appearance to a parabola, but it is not a parabola: it is a (scaled, rotated) graph of the hyperbolic cosine. The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings.The catenary is also called the alysoid, chainette, or, particularly in the material sciences, funicular.Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid, is a minimal surface, specifically a minimal surface of revolution. The mathematical properties of the catenary curve were first studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.Catenaries and related curves are used in architecture and engineering, in the design of bridges and arches, so that forces do not result in bending moments. In the offshore oil and gas industry, 'catenary' refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape.