Which of the following theorem convert line integral to a surface integral?a.Stoke's Theoremb.Mean value Theoremc.Divergence Theoremd.Rolle's Theorem

Stokes theorem gives the relation between line integral and surface integral

Suppose a vector field is given by F = (4x, −2y2, z2) and S is the surface of a cylinder definedby x2 + y2 = 4 and 0 ≤ z ≤ 3.Use a suitable volume integral given by the Divergence Theorem to show that the surfaceintegral across the total surface of the cylinder is given by 84π. You may use an appropriateJacobian without proof if stated correctly.[12 marks]2

State Fundamental theorem of Gradient, Divergence and Stokes.

Suppose F = (2x − y, yz2, y2z) is a vector field and S is the upper half surface of the spherex2 + y2 + z2 = 1, where z ≥ 0 and C is its boundary.By choosing a suitable normal, n, to the surface, S, show that (∇ × F) • n = 1. Hence showthat the surface integral over the total surface must equal π

Let I be the flux of G=<5e^y, 2xe^(x^2), 0> through the upper hemisphere S of the unit sphere.(a) Find a vector field A such that curl (A)=G.(b) Calculate the circulation of A around dS.(c) Compute I, the flux of G through S.