Consider the vector field (-7.-2.3) xr, where r= = (x,y,z). a. Compute the curl of the...
Consider the following region R and the vector field F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency. F = (3y, - 3x); R is the triangle with vertices (0,0), (1,0), and (0,2). . a. The two-dimensional curl is (Type an exact answer.) b. Set up the integral over the region R. JO dy dx 0 0 (Type exact answers.) Set up the line integral for the line...
6. Consider the vector field F = (x + sin y) î + y²z + x2 î. (a) Compute the divergence of for the point (2, -3,1). (7 points) (b) Consider F as the velocity field for fluid flow. Imagine a small drop of dye placed at the point (2, -3,1). Describe how the volume of the drop will change (instanta- neously) as the dye particles move with the flow. (3 points) (c) Compute the curl of F for the...
Consider the vector field F (x, y, z) = <y?, z2, x?>. Compute the curl (F). Use Stokes' Theorem to evaluate S. F. dr where C is the triangle (0,0,0), (1,0,0), and (0, 1, 1) oriented counter-clockwise when viewed from above.
Find the curl of the vector. Find the curl of the following vector field: t-y where b is a constant and r = x-+y +z
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
Consider the following region R and the vector field F a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in the circulation form of Green's Theorem and check for consistency. c. State whether the vector field is conservative. F-3y,3x); R is the triangle with vertices (0, 0), (1, 0), and (0, 1) a. The two-dimensional curl is D (Type an exact answer, using π as needed.) b. Set up the integral over the region R. dy...
Compute the curl of the following vector field. F = (32? sin y, 3xz? cos y, 6xz siny) The curl of Fis ( Oi+ ( Ok. +
7. Find (a) the curl and (b) the divergence of the vector field F(x, y, z)= e' sin yi+e' cos yj+zk F.de where is the curve of intersection of the plane : = 5 - x and the cylinder rº + y2 = 9. 8. Use Stokes Theorem to evaluate F(x, y, - ) = xyi +2=j+3yk
Consider a vector field given in cartesian coordinates (r, y,2) by uy (A) Calculate the curl of this vector field ▽ ˇ (B) Verify that Stokes theorem holds if the contour is the square with corners (d, d, 0), (-d, d, 0), ( d, d, 0), and (d,-d,0) aid the surface spanned by this (ont our is at 0.
Consider the vector field. F(x, y, z) = 6ex sin(y), 8ey sin(z), 5ez sin(x) Consider the vector field. F(x, y, z) = (6e* sin(y), 8ey sin(z), 5e? sin(x)) (a) Find the curl of the vector field. curl F = (-8e'sin(z), – 5e'sin(x), – 6e'sin(y)) x (b) Find the divergence of the vector field. div F = 6e sin(y) + 8e) sin(z) + 5e+sin(x)