Using Gauss's theorem, evaluate the surface integral (outward flux) of the vector field F(x, y, z)= xy cos(y)i + yz cos(z)j +xz cos(x)k across the closed surface of a cube with sides of length 7, whose one corner is at the origin, and the sides are along the 3 positive coordinate axes.

Sketch the vector field represented by F (x, y) = (-x,y)

Find all points (a, b, c) on the sphere x2 + y2 + z2 = R2 where the vector field F = yz2i + xz2j + 2xyzk is normal to the surface and F(a, b, c) ≠ 0.

Let us verify Stokes' theorem using the vector field F = (x 2 - y)i + 4zj + x 2k, where the closed contour consists of the x and y coordinate axes and that portion of the circle 2 + y 2 = a 2 that lies in the first quadrant with z = 1

Find the curl of the vector field u = a) b) ▼xu =( V. (V x u) = = (7xyz², 6x² y³, 2x³ y³ z4 ).

For the vector field (x³z², 0, -x²z), evaluate the line integral on the clase curve made by a square in the (x, 2) plane with corners at (1,0,1), (-1,01), (-1,0, -1), (1, 0,-1) in the counter clockwise direction.

a) Compute the flux of the vector field F = (-2, 3, z3³) across the disk in the xy- plane of radius 2, oriented with upward pointing normal vectors. Explain the result. b) Is the flux across the top hemisphere of the sphere x² + y² + z² 4 oriented with outward pointing normal vectors equal to the flux across the disk just computed in part a)? Explain your reasoning. -

Consider the vector field: F = (ex, In(xy), e xyz) (i) Find div(F) (ii) Find curl(F) (iii) Find div(curl(F))

Please show all of your work!

Calculate the curl of the vector g = (x−y) ax+(y+x) ay, and visualize it. Explain what does "curl of vector field" mean, in your own words. When does it equal zero?