{"id":5020,"date":"2014-03-22T06:05:03","date_gmt":"2014-03-21T21:05:03","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5020"},"modified":"2014-08-01T18:43:36","modified_gmt":"2014-08-01T09:43:36","slug":"geometric-interpretation-of-a-vector-derivative","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5020","title":{"rendered":"Geometric interpretation of a vector derivative"},"content":{"rendered":"
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If \\bold{r} is the vector joining the origin O of a coordinate system and the point (x, y, z), then specification of the vector function \\bold{r}(u) defines x, y and z as function of u. As u changes, the terminal point of \\bold{r} describes a space curve<\/em> having parametric equations x = x(u), y = y(u), z = z(u). If the parameter u is the arc length s measured from some fixed point on the curve, then <\/p>\n

\\displaystyle \\frac{d\\bold{r}}{ds} = \\bold{T}\\cdots(9)<\/p>\n

is a unit vector in the direction of the tangent to the curve and is called the unit tangent vector<\/em>. If u is the time t, then <\/p>\n

\\displaystyle \\frac{d\\bold{r}}{dt} = \\bold{v}\\cdots(10)<\/p>\n

is the velocity<\/em> with which the terminal point of \\bold{r} describes the curve. We have <\/p>\n

\\displaystyle \\bold{v} = \\frac{d\\bold{r}}{dt} = \\frac{d\\bold{r}}{ds}\\frac{ds}{dt} = \\frac{ds}{dt}\\bold{T} = v\\bold{T}\\cdots(11)<\/p>\n

from which we see that the magnitude of \\bold{v}, often called the speed<\/em>, is v = ds\/dt. Similarly, <\/p>\n

\\displaystyle \\frac{d^2\\bold{r}}{dt^2} = \\bold{a}\\cdots(12)<\/p>\n

is the acceleration<\/em> with which the terminal point of \\bold{r} describes the curve. These concepts have important applications in mechanics<\/em>. <\/p>\n<\/blockquote>\n