4. (8E given: F(x,y)- xyi-y'i Evaluate Jr where C is represented by ) as
please solve +y-1. 15.) Evaluate: F dr, where F(x, y) = xyi +(y+ x) j and C is the unit circle +y-1. 15.) Evaluate: F dr, where F(x, y) = xyi +(y+ x) j and C is the unit circle
(1 point) Suppose F(x, y) = xyi + (x – y)j and C is the triangle from (4,0) to (-4,0) to (0,4) to (4,0). (a) Find the line integral of Ể along each segment of the triangle. Along C1, the line segment from (4,0) to (-4,0), the line integral is Along C2, the line segment from (-4,0) to (0,4), the line integral is Along C3, the line segment from (0,4) to (4,0), the line integral is (b) Find the circulation...
Let F(x, y, z) = xyi+yzj + zrk and C be the boundary of the part of the paraboloid 2 =1-x2 - y2 in the first octant. The curve C is oriented anticlockwise when viewed from above. (a) Which of the following theorems most helpful in evaluating Se F. dr? (1) The Fundamental Theorem of Line Integral (2) Green's Theorem (3) Stokes' Theorem (4) Divergence Theorem (b) Let S :r(u, v) = ui + vj + (1 – 22 –...
1. Evaluate the surface Integrals using Divergence (Gauss') Theorem. a) ff(xyi +2k)ndS where S is the surface enclosing the volume in the first octant bounded by the planes z-O, y-x, y-2x, x + y+1-6 and n İs the unit outer normal to S. b) sffex.y,22)idS, where S is the surface bounding the volume defined by the surfaces z-2x2 +y, y +x2-3, z-0 and n İs the unit outer normal to S. o_ ffyi+y'j+zykids, where S is the ellipsoid.x^+-1 and iis...
5. Evaluate the integral of f along a contour y where f and y are given as follows. (a) f(x+iy) = eyel-ix along y, a positively oriented ellipse determined by the equation r = cos(20) +2. [6 (b) f(x) = 223(24 – 1)-2 along y(t) =t+iVt where 0) <t<1. [10]
(4) Evaluate the line integral F dr where C is the epicycloid with parametrization given by r(t) 5 cos t - gradient of the function f(x, y) = 3 sin(ry) + cos(y2) cos 5t and y(t) = 5 sin t - sin 5t for 0 < t < 2« and F is the (5) EvaluateF dr where F(x, y) with counterclockwise orientation (2y, xy2and C is the ellipse 4r2 9y2 36 _ F dr where F(r, y) = (x2 -...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
A B C Parametrize, but do not evaluate, //f(x, y, z) ds, where f(x, y, z) 2y22 and S is the part , where J(,y,) 3 3 and 0 Sys4 of the graph of z2 over the rectangle -2 s . Parametrize, but do not evaluate, F.n ds, where F (,-,z) and S is the sphere of radius 2 centered at the origin. Calculate JJs xyz dS where S is the part of the cone parametrized by r(u, u) (ucos...
9. Evaluate the “vector valued” line integral 1.Podr Fodr where F(x, y, z) = (x, y, zy) TT and C is given by r(t) = (sint, cost, t), with N » 4. u sta
True or False 15. If R is the disk {(x, y) 1r2+92 R, then JR f(x, y)dA 2T. 2) and f(x,y) 1 for every point (x, y) in 15. If R is the disk {(x, y) 1r2+92 R, then JR f(x, y)dA 2T. 2) and f(x,y) 1 for every point (x, y) in