SAMPLE MIDTERM 2

... (b) If f (x, y, z) = xyez and x = cos(t), y = sin(t), z = t2 , use . Express your answer as a the chain rule to compute df dt function of t. (5) Consider two parametrizations of the same curve: r(t) and r(s), where s = f (t). Show that if s0 = f (t0 ), then the vectors (T(t0 ), N(t0 ), B(t0)) = (T(s ...

... (b) If f (x, y, z) = xyez and x = cos(t), y = sin(t), z = t2 , use . Express your answer as a the chain rule to compute df dt function of t. (5) Consider two parametrizations of the same curve: r(t) and r(s), where s = f (t). Show that if s0 = f (t0 ), then the vectors (T(t0 ), N(t0 ), B(t0)) = (T(s ...

Math 106: Course Summary

... Math 106 has a lot of overlap with Math 20 (or M35), several variable calculus, but the material is covered at a more sophisticated level. You can profitably take M106 after finishing the calculus series and Math 52 (or 54). In Math 106, you apply a combination of linear algebra and several variable ...

... Math 106 has a lot of overlap with Math 20 (or M35), several variable calculus, but the material is covered at a more sophisticated level. You can profitably take M106 after finishing the calculus series and Math 52 (or 54). In Math 106, you apply a combination of linear algebra and several variable ...

Gauss` Theorem Egregium, Gauss-Bonnet etc. We know that for a

... of faces. This shows that the LHS does not depend on the particular way the surface is embedded in R3 and the RHS does not depend on the triangulation: it is a topological invariant of the surface. 4. Classification of flat surfaces Let S be a surface which is locally isometric to the plane. Gauss’ ...

... of faces. This shows that the LHS does not depend on the particular way the surface is embedded in R3 and the RHS does not depend on the triangulation: it is a topological invariant of the surface. 4. Classification of flat surfaces Let S be a surface which is locally isometric to the plane. Gauss’ ...

Lecture 8: Curved Spaces

... makes with the line of the wedge, it is different and the result is a counterclockwise angle. Thus there is a net positive curvature within that triangle. You should convince yourself that a deficit angle will be non-zero as long as the curve encloses the apex (black dot). Thus all of the curvature ...

... makes with the line of the wedge, it is different and the result is a counterclockwise angle. Thus there is a net positive curvature within that triangle. You should convince yourself that a deficit angle will be non-zero as long as the curve encloses the apex (black dot). Thus all of the curvature ...

THE FARY-MILNOR THEOREM IN HADAMARD MANIFOLDS 1

... know of only one non-integral-geometric proof to have been proposed in the intervening decades, namely that by Brickell and Hsiung [BHs]. One advantage of their proof is that it also works in hyperbolic space H 3 , although not in spaces of variable curvature. The question of whether the Fary-Milnor ...

... know of only one non-integral-geometric proof to have been proposed in the intervening decades, namely that by Brickell and Hsiung [BHs]. One advantage of their proof is that it also works in hyperbolic space H 3 , although not in spaces of variable curvature. The question of whether the Fary-Milnor ...

Lectures in Discrete Differential Geometry 3

... where M + N is the surface N ”inflated” by in the normal direction, A(M ) is the surface area of M , and H and K are the mean and Gaussian curvature, respectively. Notice that, by Gauss-Bonnet, the integral in the last term is a multiple of 4π and depends only on the genus of M . Like we did for ...

... where M + N is the surface N ”inflated” by in the normal direction, A(M ) is the surface area of M , and H and K are the mean and Gaussian curvature, respectively. Notice that, by Gauss-Bonnet, the integral in the last term is a multiple of 4π and depends only on the genus of M . Like we did for ...

On the equivalence of Alexandrov curvature and

... It is well known that the curvature bounded above (resp. below) in the sense of Alexandrov is stronger than the curvature bounded above (resp. below) in the sense of Busemann (see, e.g., [7, p. 107] or [9, p. 57]). The classical example that shows that the converse statement does not hold is the fin ...

... It is well known that the curvature bounded above (resp. below) in the sense of Alexandrov is stronger than the curvature bounded above (resp. below) in the sense of Busemann (see, e.g., [7, p. 107] or [9, p. 57]). The classical example that shows that the converse statement does not hold is the fin ...

Section 11.6

... Let x = cos t, y = sin t, and z = t. Then x2 + y 2 = cos2 t + sin2 t = 1. Thus, the curve lies on a circular cylinder of radius 1 centered about the z-axis. Since z = t, the curve moves upward along the cylinder as t increases. This curve is called a helix. ...

... Let x = cos t, y = sin t, and z = t. Then x2 + y 2 = cos2 t + sin2 t = 1. Thus, the curve lies on a circular cylinder of radius 1 centered about the z-axis. Since z = t, the curve moves upward along the cylinder as t increases. This curve is called a helix. ...

Parametric Curves

... • When we need to join two curves at a single point we can guarantee C1 continuity across the joint – Just make sure that the tangents of the two curves at the join are of equal length and direction ...

... • When we need to join two curves at a single point we can guarantee C1 continuity across the joint – Just make sure that the tangents of the two curves at the join are of equal length and direction ...

mathematics 201-nya-05

... to plot the helix r ( t ) = ( cos t ,sin t , t ) , we have > with(plots); > spacecurve([cos(t),sin(t),t],t=-Pi..5*Pi, axes=normal); You can rotate the picture with your mouse to obtain different views. For more than one plot, start by entering each curve individually (using “:” at the end not to hav ...

... to plot the helix r ( t ) = ( cos t ,sin t , t ) , we have > with(plots); > spacecurve([cos(t),sin(t),t],t=-Pi..5*Pi, axes=normal); You can rotate the picture with your mouse to obtain different views. For more than one plot, start by entering each curve individually (using “:” at the end not to hav ...

Test #2

... 15. The length l , width w and height h of a box change with time. At a certain instant the dimensions are l 2m and w h 3 and w and h are increasing at a rate of 3m / sec and l is decreasing at a rate of 1m / sec . At that instant find the rates at which the surface areas are changing. 16. The ...

... 15. The length l , width w and height h of a box change with time. At a certain instant the dimensions are l 2m and w h 3 and w and h are increasing at a rate of 3m / sec and l is decreasing at a rate of 1m / sec . At that instant find the rates at which the surface areas are changing. 16. The ...

Line and surface integrals

... Conservative vector fields The grad of every smooth scalar field is a vector field. It is natural to ask whether all vector fields are the grad of some scalar field. In general, the answer is “no”, but we can characterise those vector fields F for which this is the case. If there exists a scalar fie ...

... Conservative vector fields The grad of every smooth scalar field is a vector field. It is natural to ask whether all vector fields are the grad of some scalar field. In general, the answer is “no”, but we can characterise those vector fields F for which this is the case. If there exists a scalar fie ...

the isoperimetric problem on some singular surfaces

... the metric is continuous off a negligible set, area is lower semicontinuous (see [12, 12.5] and [6, Theorem 5.1.5]), and R is perimeter-minimizing. Regularity away from the singularities is a standard result [12, 8.6], even if M is just C 1;1 [14, Corollaries 3.7 and 3.8]. REMARK. On a 2-dimensional ...

... the metric is continuous off a negligible set, area is lower semicontinuous (see [12, 12.5] and [6, Theorem 5.1.5]), and R is perimeter-minimizing. Regularity away from the singularities is a standard result [12, 8.6], even if M is just C 1;1 [14, Corollaries 3.7 and 3.8]. REMARK. On a 2-dimensional ...

Geodesic ray transforms and tensor tomography

... Inverse problem: determine the metric g from travel times dg (x, y ) for x, y ∈ ∂M. By coordinate invariance can only recover g up to isometry. Easy counterexamples: region of low velocity, hemisphere. ...

... Inverse problem: determine the metric g from travel times dg (x, y ) for x, y ∈ ∂M. By coordinate invariance can only recover g up to isometry. Easy counterexamples: region of low velocity, hemisphere. ...

Email discussions with an Indian teacher on Mathematics, Docx

... For a simple example: y = |x| , the function is not differentiable in x = 0. (left and right derivatives are different). However for this v-shaped curve, y is obvious a minimum at x = 0. Since this problem is not too "calculus" , some books only discuss differentiable functions for max/min. For poi ...

... For a simple example: y = |x| , the function is not differentiable in x = 0. (left and right derivatives are different). However for this v-shaped curve, y is obvious a minimum at x = 0. Since this problem is not too "calculus" , some books only discuss differentiable functions for max/min. For poi ...

worksheet 4-2

... 11. The table below gives data points for a continuous function. Approximate the area under the curve on the interval [ 0, 2 ] using trapezoids and 10 equal subintervals. x f(x) ...

... 11. The table below gives data points for a continuous function. Approximate the area under the curve on the interval [ 0, 2 ] using trapezoids and 10 equal subintervals. x f(x) ...

Exotic spheres and curvature - American Mathematical Society

... 10.2A], though such objects will not concern us in this article. In dimensions one, two, and three it is well known that every topological manifold admits a unique smooth structure [FQ, 8.3]. The lowest dimension in which exotic smooth structures exist is dimension four. It is perhaps surprising tha ...

... 10.2A], though such objects will not concern us in this article. In dimensions one, two, and three it is well known that every topological manifold admits a unique smooth structure [FQ, 8.3]. The lowest dimension in which exotic smooth structures exist is dimension four. It is perhaps surprising tha ...

AN INTRODUCTION TO THE MEAN CURVATURE FLOW Contents

... on u0 . It follows that, even if you start with a heat distribution which is discontinuous, it immediately becomes smooth. Moreover, solutions converge smoothly (in C ∞ ) to constants as t → ∞ (eventual simplicity). The heat equation has some surprising properties which carry over to much more gener ...

... on u0 . It follows that, even if you start with a heat distribution which is discontinuous, it immediately becomes smooth. Moreover, solutions converge smoothly (in C ∞ ) to constants as t → ∞ (eventual simplicity). The heat equation has some surprising properties which carry over to much more gener ...

Parametric Equations and Calculus

... (b) A horizontal tangent occurs when 0 and 0 so a horizontal tangent dt dt occurs when 3t 2 3 0 which is at t 1 . When t = 1 , x = 1 and y = 2, and when t 1, x = 3 and y = 2 so a horizontal tangent occurs at the points 1, 2 and 3, 2 dy dx A vertical tangent occurs when 0 and ...

... (b) A horizontal tangent occurs when 0 and 0 so a horizontal tangent dt dt occurs when 3t 2 3 0 which is at t 1 . When t = 1 , x = 1 and y = 2, and when t 1, x = 3 and y = 2 so a horizontal tangent occurs at the points 1, 2 and 3, 2 dy dx A vertical tangent occurs when 0 and ...

Mathematics 102 — Fall 1999 Tangents

... We zoom in so that the graph of position versus time looks like a straight line. We want to know how to calculate the velocity from what we see. Now what we have seen already in class is that if the graph were a straight line, then the velocity would be equal to its slope. It is not a straight line, ...

... We zoom in so that the graph of position versus time looks like a straight line. We want to know how to calculate the velocity from what we see. Now what we have seen already in class is that if the graph were a straight line, then the velocity would be equal to its slope. It is not a straight line, ...

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold. This article deals primarily with the first concept.The canonical example of extrinsic curvature is that of a circle, which everywhere has curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.More commonly this is a scalar quantity, but one may also define a curvature vector that takes into account the direction of the bend as well as its sharpness. The curvature of more complex objects (such as surfaces or even curved n-dimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor.The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space.See the links below for further reading.