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…

a) Calculate the surface integral = ff, F.nds of the vector field: F(x, y, z) = x²yzi - xy²zj over 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 to the 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.

a) Calculate the surface integral = ff, F.nds of the vector field: F(x, y, z) = x²yzi - xy²zj over 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 to the 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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

a) Draw 4 pictures showing all possible combinations of a vector field and a curve b) Compute the circulation and the flux
Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated