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4.3 pt 1 - 4.3 Vector Fields 247 vector of the curve...

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Unformatted text preview: 4.3 Vector Fields 247 vector of the curve coincides with the vector field, as in Figure 4.3.9. Flow lines provide a geometric picture of a vector field, enabling us to understand it more fully. A flow line may be viewed as a solution of a system of differential equations. Indeed, we can write the definition c’{t) = F(c(t)) as 3’05) = P(w(t).y(t],z(t))s 9’0!) Qlwitllamidtlla fit) = Rixltlwmlzttlli where F = Pi + Qi + Rk. One learns about such systems in courses on differential equations. Show that the path C(t] = {cos t, sin t) is a flow line of F(:i:, y) = —yi+a:j. Can you find others? Solution We must verify that c’[t) = F(c(t)). The left side is (—sin t)i + (cos t)i while the right side is F( cos t, sint) = {— sin tJi-l- ( cos t)i, so we have a flow line. As suggested by Figure 4.5.5, the other flow lines are also circles. The}r have the form C(t) = (r cos {t — ta), r sin (t — to)) for constants r and t0. 0 In many cases explicit formulas for flow lines are not available, so one must resort to numerical methods. Figure 4.5.10 shows some output from a program that computes flow lines numerically and plots them on the screen. In Exercises 1-8, sketch the given vector field or a small multiple of it. 1. F(a:,y) = (2, 2) 2. F(a:,y) = (4,0) 3. FLT, y) = (I, a} 6- 1:63:90 : (yr—2x) ...
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