B.2 Stokes' Theorem [5] a. Give Stokes theorem, defining all quantities involved, (i) in three dimensions in its standar

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B 2 Stokes Theorem 5 A Give Stokes Theorem Defining All Quantities Involved I In Three Dimensions In Its Standar 1 (37.64 KiB) Viewed 39 times

B.2 Stokes' Theorem [5] a. Give Stokes theorem, defining all quantities involved, (i) in three dimensions in its standard integral form (draw a figure), and clarify how the orientation problem is resolved. (ii) in one dimension b. Consider the following coordinates [5] Derive the Laplacian in these coordinates by using the formula from the course. Consider the function / =ys and evaluate / in both Cartesian and the coordi- nates defined above. c. Consider the following vector field [5] A=677 + m2 *) 22+ in Cartesian coordinates. Compute the curl of this vector field and integrate its flux through the surface bounded by the upper hemisphere S lg = / x 4.4 , [5] d. Compute the line integral le = lidi, Sead where is the boundary of the surface S. Compare the two results obtained and explain why Ic # Is (1.0. Stokes' theorem does not hold here). Describe how to amend S for Stoken' theorem to hold. 15)