Proof of Stokes’ Theorem Confirm the following step in the proof of Stokes’ Theorem. If z = s(x, y) and f, g, and h are functions of x, y, and z, with M = f + hz_x and N = g + hz_y, then M_y = ƒ_y + ƒ_z z_y + hz_(xy) + z_x(hy + h_z z_y) and N_x = g_x + g_z z_x + hz_(yx) + z_y(h_x + h_z z_x).

FizeauV 2021-01-27 Answered
Proof of Stokes’ Theorem Confirm the following step in the proof of Stokes’ Theorem. If z = s(x, y) and f, g, and h are functions of x, y, and z, with M = f + hz_x and N=g+hzy, then
My=ƒy+ƒzzy+hzxy+zx(hy+hzzy)
and
Nx=gx+gzzx+hzyx+zy(hx+hzzx).
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Expert Answer

lamanocornudaW
Answered 2021-01-28 Author has 85 answers
Step 1
Considering the steps of proof of stokes theorem,
if, z = s(x,y)and f(x,y,z),g(x,y,z),h(x,y,z)
Hence the partial differentiation(using the chain rule) is,
My=(f(x,y,z)+hzx)y
=fy+fzz(x,y)dy+hzxy+zx(hy)+hzz(x,y)y
=fy+dzzy+hzxy+zx(hy+hzzy)
Step 2
Further,
Nx=(g(x,y,z)+hzy)x
=gx+gxz(x,y)x+hzyx+zy(hx+hz(dzx,ydx))
=gx+gzzx+hzyx+zy(hx+hzzx)
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