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Curves

A curve is representative of a line which is not straight. A curve begins and finishes similar to a line, but stops resembling a straight line as it approaches the formation of its bulge or deformation. For instance, a parabola is an example of a curve. 

Figure 1. This illustration is representative of a curve. As is depicted in this figure, a curve bulges outwards near the middle.

Curves are generally thought to be plane curves and are therefore two dimensional. However, in some cases curves can be three dimensional. These types of curves are referred to as space curves.

In the study of geometry, curves are particularly important to the areas of differential geometry and algebraic geometry. In differential geometry, a curve is referenced as a differentiable curve and this curve does not exist in two dimensional space.

In algebraic geometry, curves are known as algebraic curves and can be either plane curves, space curves or of a higher dimension. For example, an elliptic curve, which is studied in number theory, is an example of an algebraic curve. 

Curve Fitting and Input and Output Files

Problem 2 This is a three dimensional version of the two dimensional curve fitting problem associated with determining thevertical alignment of a road. In this case, the problem is that groundwater hydrologists want to map thepiezometric surface (top of the water table) in a region so they can assess the potential for developmen

Gaussian Curvature of the Unit Sphere

Compute the curvature of the unit sphere in R^3. I started the problem with r = [sin u cos v sin u sin v cos v. ] From there, how do you get the principal curvatures k_1 and k_2?

Area under a Curve

Calculate the area under the curve y=1/(x^2) above the x-axis on the interval [1, positive infinity]. keywords: finding, find, calculate, calculating, determine, determining, verify, verifying

Finding Curvature

Compute the curvature k(t) of the curve r(t) = 2t i + 4sint j +4cost k

Area under the curve

Find the area of the surface obtained when the graph of y=x^2 , 0<= x <= 1, is rotated around the y axis.

Calculating Volumes of Bounded Regions (Curve, Origin and Interval)

1. Sketch the region bounded between the given curves and then find the area of each region a) {see attachment} b) Find the area of the region than contains the origin and is bounded by the line {see attachment} 2. Sketch the given region and then find the volume of the solid whose base is the given region and which has the

Intersection of Curves

Find the points of intersection (if any) of the given pair of curves and draw the graphs. Y= x^2 and y= 2x+2

Example of a non-rectifiable closed Jordan curve.

Give an example of a non-rectifiable closed Jordan curve on the interval -1<=t<=1. My thought: t + i(sin 1/t) + ????? Please advise what curve I can add to make this work. Or, if this will not work, please provide an example of a non-rectifiable closed Jordan curve on -1<=t<1.

Length of a arc

Two people 1.8m tall walk from each other until they can no longer see each other (due to the curvature of the earth which has a radius of 6378km). Assuming nothing else blocks their view, how far do they have to walk? Note. I cant get my head around how this relates to what we've lear

Curvature of the curve and torsion of the curve

Differential Geometry (I) Curves in Space Curvature of the Curve Torsion of the Curve For the curve r = ( &#8730;6 at^3, a(1+3t^2), &#8730;6 at ) Show that k = - T = 1/[a(3t^2 + 1)^2] where k = curvature of the curve, T = tors