a) Calculate the surface integral= ff, F.nds of the vector field:F(x, y, z) = x²yzi - xy²zjover the surface S of the unit cube defined by the intersection of the planes x = 0,x = 1, y = 0, y = 1, z = 0 and z = 1, where n denotes the unit vector normal tothe surface element dS pointing in the outward direction.b) Using Gauss's (divergence) theorem re-evaluate the surface integral in part a)above by means of a volume (triple) integral and comment on the result.c) It can be shown that a vector field is conservative if and only if it can be writtenas the gradient of a scalar field termed "potential function". By constructing such apotential function, show that the vector field:F(x, y, z) = 2xyl + (x² + 2yz)j + y²kis conservative.d) Calculate via direct integration the line integralfF.dlwhere F is defined in part c) and the path of integration is the straight line connectingpoints (0,0,0) and (1,1,1). Verify your answer by using the potential function you haveconstructed in part c).Proctortrackm Paper...

We will draw figure to find normal vectors of all sides and find surface integral . At end add all…

Answered: a) Calculate the surface integral = ff,…

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Vectors In Two And Three Dimensions

Section9.6: Equations Of Lines And Planes

Problem 2E

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