Please show all work Thanks Let a be the disk x y z x l2 l

Please show all work. Thanks.

Let a be the disk {(x, y, z) | (x- l)^2 + lessthanorequalto 9, y = 17}, and let C be the boundary of a oriented counterclockwise if viewed from the \"top\" of the positive y-axis. Suppose F(x, y. z) = f(x, y, z)i + g(x, y, z)j + h(x, y, z)k, where f, g, and h have continuous first partials. Also assume that f_z = h_x throughout some open set containing sigma. Prove that F T ds = 0.

Solution

Solution: Given

F(x,y,z) = f(x,y,z)i + g(x,y,z) j + h(x,y,z)k , f_z = h_x

Now

curl F = i(\\frac{\\partial h}{\\partial y} - \\frac{\\partial g}{\\partial z})

+ j(\\frac{\\partial f}{\\partial z} - \\frac{\\partial h}{\\partial x})

+k(\\frac{\\partial g}{\\partial x} - \\frac{\\partial f}{\\partial y})

=i(h_y - g_z) + j(f_z - h_x) + k(g_x -f_y)

= i(h_y - g_z) + j.0 + k(g_x -f_y) (as f_z = h_x)

=i(h_y - g_z) + k(g_x -f_y)

Here given C is in xz-plane and y-axis is perpendicular to C.

So in this case normal n is j along y-axis.

So n. curl F = j.[i(h_y - g_z) + k(g_x -f_y)] = (j.i)(h_y - g_z) + (j.k)(g_x -f_y)

= 0.(h_y - g_z) + 0.(g_x -f_y)] ( j.i =j.k =0)

=0 .........................................(1)

Also

\\smalloint_{C} \\bf{F.T}ds = \\smalloint_{C} \\bf{F.\\frac{dr}{ds}}ds

=\\smalloint_{C} \\bf{F.dr}........................(2)

Hence by Stoke\'s theorem

\\smalloint_{C} \\bf{F.dr} =

int_{C} \\bf{F.dr} = \\int\\int_{S}(curl F). ndS = 0 ( by equation (1))

Hence the required line integral is zero.(proved)