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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?

Trends, Forecasting and Curve-Fitting

Picture this- 2001----------------128 2002-----------------192 2003----------------288 2004-----------------432 2005-----------------648 How many catfish will there be in 2006? Please explain and give me a strategy for solving this problem?

Tangent Curves

Y=alpha*x^2 + alpha*x + (1/24) x=alpha*y^2 + alpha*y + (1/24) These are two curves. For what values of alpha are these two curves tangent to each other.

Contact of Sphere and a Sphere Surface

A spacer defines the air-gap distance between a lens and a flat surface. The corner of the spacer can be modeled as a circle with radius, r1. First a circular lens is used, and the "sag," or amount that the lens bulges out on the axis, defines the air gap distance by: sag = R - (R2 - (chord)2) ½ In this case the contact p

Contact Point of Two Circles

If two circles with radii r and R contact at a point above a straight axis, what is the height, Δy, of contact (as a function of the curvature of the circles) above the axis. Please see the attached file for the fully formatted problem.

Arc Length

Find the arc length: x^2 = (y-1)^3 on [0,8] 0<=x<=8

Sketch Region

Prob: Sketch the region enclosed between 2 - z = x2 + y2 and z2 = x2 + y2. Describe their curve of intersection

Curvature

Find k(t) for x = e^(3t) , y = e^(-t) .

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.

Curvature of a curve in space

The curvature of a curve in space r(t) is given by k(t) = | r'(t) à? r''(t) | / | r'(t) |^3 . Consider now the curve r(u) = r(sigma(u)), given by the reparametrization t = sigma(u) of the initial curve. Show that the curvature k of the curve r is given by k(u) = k(sigma(u)), where k is the curvature of the initia

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

Question about Curve Sketching Step-by-Step

I need help using the guidelines of curve sketching to sketch y=(x^2)/((x^2)+3). The steps seem more complicated than they should be to me, and I can't seem to get anything remotely looking like a correct answer.

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.