Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

The solution is an application of Stokes' theorem. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Given the vector field F=3yi + (5-2x)j + ((z^2)-2)k find
a) Div F
b) Curl F
c) The surface integral of the normal component of curl F over the open hemispherical surface (x^2)+(y^2)+(z^2)=4 above the xy plane.
* Hint: by a double application of Stokes theorem, part c can be reduced to a triviality

Stokes Theorem. See attached file for full problem description.
1. compute the line integral where F = (yz^2 - y)i + (xz^2 + x)j + 2xyzk where C is the circle of radius 3 in the xy-plane, centered at the origin, oriented counterclockwise as viewed from the positive z -axis.
2. Given F =yi - xj + yzk and the region S determ

See attached file for full problem description.
Use Stokes'theorem to evaluate the surface integral of the curl:
where the vector field F(x,y,z) = -12yzi + 12xzj + 18(x^2+y^2)zk and S is the part of the paraboloid z = x^2 + y^2 that lies inside the cylinder x^2 + y^2 =1, oriented upward.

1. A vector ﬁeld v(x, y, z) is given by the formula
v(x, y, z) = xyˆx − yˆ2y.
Consider a square path in the xy plane which starts at (0,0,0) and moves along the corners (1,0,0), (1,1,0) and (0,1,0). Calculate the path integral of v, i.e. v · dr, and calculate the area integral of the divergence, R ∇ × v · da, an

Prove that if D is the closed disc |x| 1 in R2, then any map f 2 C2[D ! D] has a fixed point: f(x) = x. The proof is by contradiction, and uses Stokes theorem. Follow the steps outlined below.
(1) Define a new map F(x) = 1
....
Show that F has no fixed points if r is small enough.
(2) Draw the ray from F(x) to x (these ar

Describe the additional assumptions , conditions ,and steps required to derive the Navier-Stokes Equations from the momentum equations. Use as little mathematics as possible.

Using Green's Theorem and Stokes'Theorem respectively, calculate the given line integrals.
• Using Green's Theorem calculate the line integral , where along the positively oriented closed curve C which is the boundary of the domain: .
Which line integrals you would have to evaluate instead in order to calculate h