Begin2.DVI
This is interpreted as showing the vector ˆet rotates about a line through eˆb with angular velocity ω. If the curvature κ= 0 , ...
Example 7-5. Determine how to calculate the torsion τ of a space curve. Solution The derivative dr ds = ˆet is a unit tangent v ...
Analysis of this equation demonstrates that the velocity vector v is directed along the tangent vector at any time t and has th ...
To represent a curve on a given surface defined in terms of two parameters uand v, one can specify how these parameters change. ...
97 Coordinate curves on sphere. For example, consider the unit sphere r (u, v ) = cos usin vˆe 1 + sin usin vˆe 2 + cos vˆe 3 w ...
Figure 7-4. Sphere centered at origin and projections onto planes x= 0,y= 0 and z= 0. If α (^2) +β (^2) +γ 2 4 −δ= ...
latitude on the sphere given by λ=π 2 −θand when φis held constant, one obtains a coordinate curve representing some line of lon ...
Figure 7-6. Elliptic paraboloid The Elliptic Paraboloid The elliptic paraboloid centered at the point (x 0 , y 0 , z 0 )is descr ...
The position vector r =r (u, v ) = au cos vˆe 1 +bu sin vˆe 2 +cu ˆe 3 describes a point on the surface centered at the origin ...
It can also be represented by the parametric equations x−x 0 =acos vsinh u, y −y 0 =bsin vsinh u, z −z 0 =ccosh u (7 .37) where ...
The Hyperbolic Paraboloid The hyperbolic paraboloid centered at the the point (x 0 , y 0 , z 0 ) is described by the equation z− ...
axis. A paraboloid is obtained by rotating the parabola y=x^2 , 0 ≤x≤x 0 about the x= 0 axis. The general procedure for determin ...
Solution In the figure 7-10 the point Prepresents a general point on the space curve. Let the coordinates of this point be denot ...
the magnitude of the vector r P−r Qand one can determine this distance from the dot product relation R^2 = (r P−r Q)·(r P− ...
Next construct the plane which passes through the point P and is perpendicular to the axis of rotation. The equation of this pla ...
In general, a ruled surface can be thought of as set of points created by moving a straight line. One way of creating the equati ...
Surface Area The position vector r =r (u, v ) = x(u, v )ˆe 1 +y(u, v )ˆe 2 +z(u, v )ˆe 3 α≤u≤β, γ ≤v≤δ (7 .51) defines a surfa ...
of the cross product of the sides of a parallelogram gives the area of the parallelogram and consequently one can express the el ...
which represents a summation of the elements of surface area over the surface be- tween appropriate limits assigned to the param ...
Arc Length Consider a curve u=u(t),v=v(t) on a surface r =r (u, v )for t 0 ≤t≤t 1. The element of arc length ds associated wit ...
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