Find parameterisations x(t), y(t), t є 1 (a) (b) (c) for each of the following curves: 2. The straight line from 4+2i to 3+5i; The circle of radius 2 centered at 3-i oriented counter-clockwise; The g...
6. Estimates from geometric definitiions: (a) Suppose divF2z. Estimate the ux of F through a sphere of radius 0.01 centered at (b) Suppose curlF-(z + 4)it (2-ทั+ (:-3)E, estimate circulation of F around a circle C of radius 0.1, centered at the origin, if C is on the ry- yz-, and rz-plane respectively oriented counter-clockwise when viewed from the positive z, positive a, and positive y-axis respectively 7. Three small squares Ci,C2, Cs each with sides 0.1, centered at the...
10. Let F(x, y, z) = 〈y,-z, 10) per half of x2 +y2 + z2 = 1, oriented upward, and C the circle 2 y 1 in the z - y plane, oriented counter-clockwise. Find Jscurl(F) ndS directly and by using Stokes' Theorem. , where S is the up 10. Let F(x, y, z) = 〈y,-z, 10) per half of x2 +y2 + z2 = 1, oriented upward, and C the circle 2 y 1 in the z - y...
6. Consider a triangle with vertices at 1, 2 + 2i, 3-i oriented clockwise. (a) Draw the triangle and mark the orientation on its edges. (b) Find a parametrization for each of its edges 6. Note that parametrization of a straight path from ะเ to 22 is:(t)- + t(22-a), 0 1. t
7. Use Green's Theorem to find Jc F.nds, where C is the boundary of the region bounded by y = 4-x2 and y = 0, oriented counter-clockwise and F(x,y) = (y,-3z). what about if F(r, y) (2,3)? x2 + y2 that lies inside x2 + y2-1. Find the surface area of this 8. Consider the part of z surface. 9. Use Green's Theorem to find Find J F Tds, where F(x, y) (ry,e"), and C consists of the line segment...
Find a parametrization of the circle of radius 4 in the xy-plane, centered at (−2,−4), oriented counterclockwise. The point (2,−4) should correspond to t=0. Use t as the parameter for all of your answers.
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Compute wise (Hint: use the FT of line integrals. We could not use it for the circle centered at the origin, but we can use the theorem for this circle. Why?) (b) Let 0 be the angle in polar coordinates for a point (x, y). Check that 0 is a potential function for F 3. Consider the...
Determine the vector-function for an ellipse whose center is (2, −3), has an x-radius of 2, a y-radius of 4, starts at the point (2, −7), and travels in the counter-clockwise direction. What is the rectangular equation for this ellipse? Sketch the ellipse
Select the first set of parametric equations, x = a cos(bt), y = c sin(dt). (a) Set the equations to x = 2 cos(t), y = 2 sin(t) using the sliders for a, b, c, and d. Describe the parametric curve. This answer has not been graded yet. What minimum parameter domain is required to draw the entire circle? Osts How many times is the circle traced out for Osts 4? Click the Animate button and observe the relationship between...
Give parametric equations that describe a full circle of radius R, centered at the origin with clockwise orientation, where the parameter t varies over the interval [0,22]. Assume that the circle starts at the point (R,0) along the x-axis. Consider the following parametric equations, x=−t+7, y=−3t−3; minus−5less than or equals≤tless than or equals≤5. Complete parts (a) through (d) below. Consider the following parametric equation. a.Eliminate the parameter to obtain an equation in x and y. b.Describe the curve and indicate...
True or False Determine whet her the statement is true or false, and circle the correct answer. Each question is worth 2 points. (1) If F is a vector field and C is an oriented curve, then F dr must be less than zero. F (2) It is possible that for a certain vector field F and piecewise smooth oriented path C we have/. F. dr-2i-Sj. (3) Suppose d·is the unit square joining the points (0,0), (1,0), (1,1), (0.1) oriented...