4100 AWL/Thomas_ch13p906-964
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چکیده
As a particle moves along a smooth curve in the plane, turns as the curve bends. Since T is a unit vector, its length remains constant and only its direction changes as the particle moves along the curve. The rate at which T turns per unit of length along the curve is called the curvature (Figure 13.19). The traditional symbol for the curvature function is the Greek letter (“kappa”). k T = dr>ds 13.4
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4100 AWL/Thomas_ch13p906-964
If you are traveling along a space curve, the Cartesian i, j, and k coordinate system for representing the vectors describing your motion are not truly relevant to you. What is meaningful instead are the vectors representative of your forward direction (the unit tangent vector T), the direction in which your path is turning (the unit normal vector N), and the tendency of your motion to “twist” ...
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OVERVIEW When a body (or object) travels through space, the equations and that give the body’s coordinates as functions of time serve as parametric equations for the body’s motion and path. With vector notation, we can condense these into a single equation that gives the body’s position as a vector function of time. For an object moving in the xy-plane, the component function h(t) is zero for a...
متن کامل4100 AWL/Thomas_ch16p1143-1228
We know how to integrate a function over a flat region in a plane, but what if the function is defined over a curved surface? To evaluate one of these so-called surface integrals, we rewrite it as a double integral over a region in a coordinate plane beneath the surface (Figure 16.38). Surface integrals are used to compute quantities such as the flow of liquid across a membrane or the upward fo...
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In the calculus of functions of a single variable, we used our knowledge of lines to study curves in the plane. We investigated tangents and found that, when highly magnified, differentiable curves were effectively linear. To study the calculus of functions of more than one variable in the next chapter, we start with planes and use our knowledge of planes to study the surfaces that are the grap...
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