A vector field is specified by A(x, y, z) = aye_x - bxe_y. Show that A(x,...
a) A vector field F is called incompressible if div F = 0. Show
that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is
incompressible.
b) Suppose that S is a closed surface (a boundary of a solid in
three dimensional space) and that F is an incompressible vector
field. Show that the flux of F through S is 0.
c)Show that if f and g are defined on R3 and C is a closed curve
in R3 then...
(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 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)
Consider the vector field. F(x, y, z) = (3ex sin(y), 3ey sin(z), 5e7 sin(x)) (a) Find the curl of the vector field. curl F = (-3d"cos(z))i – (36*cos(x)); – (5e+cos(y) )* * (b) Find the divergence of the vector field. div F = 3e'sin(y) + 3e'sin(z) + 5e+ sin(x)
4. If V(x,y,z)-6xy2xyz -3xy'z, Find the value of the electric field (in vector notation) at the point (3,-
Consider the vector field. F(x, y, z) = (98 sin(y), 4e' sin(z), 2e sin(x)) (a) Find the curl of the vector field. curl F = (b) Find the divergence of the vector field. div F =
Sketch by hand the 3D vector field F(x, y, z)- -yk. Label everything.
Sketch by hand the 3D vector field F(x, y, z)- -yk. Label everything.
Previous Problem Problem List Next Problem (1 point) Show that the vector field F(x, y, z) show what you intended? (-3y cos(5x), 5x sin(-3y),0) is not a gradient vector field by computing its curl. How does this curl(F) = V × F-《
Let F(x, y, z) be the gradient vector field of f(x, y, z) = exyz , let C be the curve of the intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1, oriented counterclockwise, evaluate Sc F. dr. OT O -TT O None of the above. 00
3. A vector field is given by F(x,y,z) y'zi - x3j+ xyk. Calculate curl(F). Show all working. 4 marks)