1.34. Draw the path parametrized by r) Cos(t)cos(t)+i sin(t ) sin(t 0sIS2 1.31. Show that the...
do the three with mark 1.31. Show that the union of two regions with nonempty intersection is itself a region. 1.32. Show that if ACB and B is closed, then dAC B. Similarly, if AC B and A is open, show that A is contained in the interior of B 1.33. Find a parametrization for each of the following paths: (a) the circle C[1+ i, 1], oriented counter-clockwise (b) the line segment from -1-to 2 (c) the top half of...
Problem 2. Let a and b be constants. For the parametrized curve R(t) = (eat cos bt, eat sin bt), find the angle between R(t) and the tangent vector at R(t).
Calculate Tr, T. and N(r, θ) for the parametrized surface at the given point. | I θ . r ., G(r, θ)-(r cos(9), r sin(θ), 1-r2); 16' 4 6' 4 6' 4 Find the equation of the tangent plane to the surface at that point. Calculate Tr, T. and N(r, θ) for the parametrized surface at the given point. | I θ . r ., G(r, θ)-(r cos(9), r sin(θ), 1-r2); 16' 4 6' 4 6' 4 Find the equation...
2 (7 points each) Consider the circle parametrized by r(t) 3,6 cos t, 6 sin t). (a) Compute its are length over the interval 0 < wfind an are leugth pi of the circle. 2 (7 points each) Consider the circle parametrized by r(t) 3,6 cos t, 6 sin t). (a) Compute its are length over the interval 0
QUESTION 4 Suppose a fourth field and path: F= <cos(z), sin(z), xy > and r= <sin(t), cos(t), t-> when Osts 21 What does this field look like? What does the path look like? Find ff. dr (use a calculator), what does it represent? Explain.
1L COS v 21) Let H denote the surface parametrized by r(u, )sin, where 7 0S11 land 0 < u < 2T. (a) Compute Tu, Tu, and Tu X T, (b) Compute 1L COS v 21) Let H denote the surface parametrized by r(u, )sin, where 7 0S11 land 0
2. A dragon is flying around in a pattern given by the parametric curve r(t) (cos(t) cos((sin(t) sin(t) cos(t)j. cos(t) - cos sin(t)-sin(t) cos(t))j (a) Find a formula for the velocity of the dragon at time t (b) Find all the times at which the dragon's speed is zero. Explain your reasoning. c) Does the path of the dragon contain any cusps? Explain your reasoning 2. A dragon is flying around in a pattern given by the parametric curve r(t)...
Problem 1. Let y be the segment [0, 2] C C parametrized by r(t) = tz, te[0,1] C R. Compute the path integral ew dw. Problem 2. Let 7 be the path defined by (O) = ei0, 0 (0,21] Compute the integral sill sin w dw. w
Problem 4.9 (e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
Problem 2. Consider the two parametrized curves r(t) = (1+,2-t,t + 382 – 4t + 4) and r(u) = (u?, 3 - u, u' + 22 - 6u + 8), where t and u are in R. (a) Find the coordinates of the point of intersection P of the two curves. (b) The curves traced out by ry and r2 lie on a surface S. Find an equation of the tangent plane to the surface S at the point P...