4. Let C be the boundary of the quadrilateral with vertices (1, 1), (1,2), (2, 3)...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2, 3) and (2, 1), oriented positively i.e. counterclockwise). Let F be the vector field F(1, y) = (e* + y²)i + (ry + cos y)j. Compute the line integral F. dr. 10. (15 Points) Let S be the portion of the paraboloid z = 1-rº-ythat lies on and above the plane z = 0. S is oriented by the normal directed upwards. If F...
In this problem, let li be the line that passes through the points A(1,2, 4) and B(-1,3,8), and let l2 be the line with symmetric equations x +1 = 2y = 32 — 3. Parts (e) and (f) relate to the vector field F = (xy, xz, yz). (a) Show that the lines li and l2 intersect. (b) Let P be the plane that contains both lines li and lz. Find an equation for P. (c) Show that the points...
Let F(x, y,z) = < x + y2,y + z2,z + x2 >, let S be a surface with boundary C. C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1). 8. a. Evaluate F dr curl F ds b. Let F(x, y,z) = , let S be a surface with boundary C. C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1). 8. a. Evaluate F dr curl F ds b.
DETAILS 3. [2/4 Points) Consider the given vector field. F(x, y, z) = (e", ely, exy?) (a) Find the curl of the vector field. - yzelyz lazenz curl Fe (b) Find the divergence of the vector field. div F = ertxely tuxely F. dr This question has several pa You will use Stokes' Theorem to rewrite the integral and C is the boundary of the plane 5x+3y +z = 1 in the fir F-(1,2-2, 2-3v7) oriented counterclockwise as viewed from...
(1 point) Let +4z + 4 sin (a) Find curl F. curl F- (b) What does your answer to part (a) tell you about JcF dr where C is the circle (x 30)2 + (y - 10)2 1 in the xy-plane, oriented clockwise? (e) If C is any closed curve, what can you say about fcFdr? (d) Now let C be the half circle (-30)2-cy-10)2-1 in the xy-plane with y 10, traversed from (31, 10) to (29, 10). Find F...
1. Let F(x, y, z) = (-y + ,2-2,2-y), and let S be the surface of the paraboloid 2 = 9-32 - v2 for 2 > 0. oriented by an upward pointing normal vector. Note that the boundary of S is C, the circle of radius 3 in the xy-plane. Verify Stokes' Theorem by computing both sides of the equality: (a) (1 Credit) || (D x F). ds (b) (1 Credit) $F. dr
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove İ}!").lx (ly İ)(2 dr dy. Pdr+Qdy That is, prove Green's Theorem for the triangle A. [Hint: the lecture notes have a proof for when A is a rectangle. So, the idea is is to give a similar proof where we have this...
y? +1 e a vector field 12) Let F(x,y) = (10 + 15y3 + cos (In(xe*)))i + (-6- 15x3 – sin" (ev? V In on R2. Use Green's Theorem to compute (F. dr Where C is the negatively oriented boundary of the region bounded by x2 + y2 = 4 and x2 + y2 = 9, in the first quadrant only.
(10 pts) Evaluate where C is the boundary of the square with vertices (0,0), (1,0),(0,1) and (1, 1) oriented clockwise. (10 pts) Evaluate where C is the boundary of the square with vertices (0,0), (1,0),(0,1) and (1, 1) oriented clockwise.
need 1-5 Midterm #3, Math 228 Each question is worth five points. 1. Let F(r.yzy). Let C be any curve that goes from A(-1,3,9) to B(1,6,-4). a) Show that F is conservative. b) Find a function φ such that ▽φ = F c) Use the result of b) to find Ic F Tds 2. Let F(z, y)-(2), and let C be the boundary of the square with vertices (1, 1). (-1,1). (-1,-1 traced out in the counter-clockwise direction. Find Jc...