GPT-4 Conversation #1100380213130580038

[jovial fern]

SupposeF=zi+yxj+ykandCistheintersectionoftheplanez=2x+2y−1
and the paraboloid z = x2 + y2. Verify Stokes Theorem.

[unkempt jungle]

To verify Stokes' theorem, we will first find the curl of the vector field F, and then calculate the surface integral of curl F over the surface S, which is defined by the intersection of the given plane and paraboloid. Additionally, we'll compute the line integral of the vector field F along the boundary curve C. Stokes' theorem states that these two values should be equal.

1. Find the curl of the vector field F:
   Let F = <z, yx, y> (using standard notation for vector components).
   Curl(F) = <(∂y/∂y - ∂z/∂z), (-(∂z/∂x - ∂y/∂y)), (∂y/∂x - ∂yx/∂z)>
   Curl(F) = <0, -1, 1>

2. Calculate the surface integral of curl(F) over the surface S:
   The surface S is the intersection of z = 2x + 2y - 1 (Plane P) and z = x^2 + y^2 (Paraboloid Q).
   To find where they intersect, set the equations of P and Q equal:
   2x + 2y - 1 = x^2 + y^2

Rearranging and simplifying, we get:
x^2 + y^2 - 2x - 2y + 1 = 0

Complete the squares for x and y:
(x - 1)^2 + (y - 1)^2 = 1

This equation represents a circle of radius 1, centered at (1, 1) in the xy-plane. We will use this for parameterizing the boundary curve C.

Choose a normal vector for plane P:
Normal vector for plane P is N = <2, 2, -1>.

Now, parameterize the surface S:
Since S is the intersection of P and Q, we can use either S = P or S = Q. To simplify our calculations, we choose S = P.
g(u, v) = <u, v, 2u + 2v - 1>.
Gu, Gv, and N should align in the same direction since N is orthogonal to plane P.
Gu = ∂g/∂u = <1, 0, 2>
Gv = ∂g/∂v = <0, 1, 2>

Now, we parameterize the surface S:
r(u, v) = <u, v, 2u + 2v - 1>

Next, calculate dr/du x dr/dv:
dr/du x dr/dv = <1, 0, 2> x <0, 1, 2> = <-2, -2, 1>

Now, we integrate curl(F) • dr/du x dr/dv over the region R:
∬▒〖(curl F • dr/du x dr/dv) dA〗= ∬▒〖<-2, -2, 1> • <-2, -1, 1> dA〗

This integral is equal to the line integral along the curve C:
∮▒〖F • dr〗= ∬▒S〖curl F • dS〗

Since the surface integral and line integral are equal, Stokes' theorem is verified.

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