Calculating doubleintM (∇×F) ·dS

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In summary: That's where the "ds" comes from.In summary, in order to solve the problem of computing doubleintM (∇×F) ·dS, the integral should be set up using the line integral with respect to the bottom circle of the capped cylindrical surface M, which is parameterized as (3cosθ, 3sinθ, 0). The vector field F = (zx + z^2y + 2y, z^3yx + 8x, z^4x) should be used, and with z = 0, it becomes (2y, 8x, 0). The integral should be set up with limits of 0 to 2pi, and the vector
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Borat321
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Homework Statement


Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 9, 0 ≤ z ≤ 1, and a hemispherical cap defined by x^2 + y^2 + (z−1)^2 = 9, z ≥ 1. For the vector field F = (zx + z^2y + 2 y, z^3yx+ 8 x, z^4x^2), compute doubleintM (∇×F) ·dS in any way you like.

doubleintM (∇×F) ·dS = ?


Homework Equations



I thought the line integral would make the most sense in solving this - I wanted to take the line integral with respect to the bottom circle. parameterized, it is (3costheta, 3sintheta, 1).

Line int = F(r) (rprime)

The Attempt at a Solution



I have no idea what's going on here - I tried to take the line integral by saying r=(3sint,3cost,0) Since z=0 for circle the only parts that matter in the F is the 2y and the 8x, but I don't think that's right.

Integral w/ limits 0 to 2pi

(2(3sintheta),8(3costheta),0) dot product (-3sintheta, 3costheta, 0)

Can anyone help me with this one?
 
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  • #2
Borat321 said:

Homework Statement


Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2 + y^2 = 9, 0 ≤ z ≤ 1, and a hemispherical cap defined by x^2 + y^2 + (z−1)^2 = 9, z ≥ 1. For the vector field F = (zx + z^2y + 2 y, z^3yx+ 8 x, z^4x^2), compute doubleintM (∇×F) ·dS in any way you like.

doubleintM (∇×F) ·dS = ?


Homework Equations



I thought the line integral would make the most sense in solving this - I wanted to take the line integral with respect to the bottom circle. parameterized, it is (3costheta, 3sintheta, 1).
Not quite. That is the bottom of the hemispherical cap. The bottom of M is the circle in the xy-plane: [itex](3 cos\theta , 3 sin\theta ,0)[/itex]

Line int = F(r) (rprime)

The Attempt at a Solution



I have no idea what's going on here - I tried to take the line integral by saying r=(3sint,3cost,0) Since z=0 for circle the only parts that matter in the F is the 2y and the 8x, but I don't think that's right.
Oh! Okay, now you moved down to the correct circle!

Integral w/ limits 0 to 2pi

(2(3sintheta),8(3costheta),0) dot product (-3sintheta, 3costheta, 0)

Can anyone help me with this one?
[itex]F= (zx+ z^2y+ 2y, z^3yx+8x, z^4x)[/itex] which, on z= 0 becomes
[itex]F= (2y, 8x, 0)[/itex]. Good. That's what you have. But [itex]s= (3 cos(\theta), 3 sin(\theta), 0)[/itex] so [itex]ds= (-3 sin(\theta), 3 cos(\theta), 0)d\theta[/itex]. Your integral should be:
[tex]\int_{\theta= 0}^{2\pi}(6sin(\theta ), 24cos(\theta ),0)\cdot (-3sin(\theta ),3cos(\theta ),0)d\theta[/tex].
Never forget the "d" in an integral!
 

What is the meaning of the double integral in "Calculating doubleintM (∇×F) ·dS"?

The double integral in this context represents the integration of a vector field over a two-dimensional surface, which is often used in the study of fluid dynamics and electromagnetism.

What does the ∇ symbol represent in this equation?

The ∇ symbol, also known as the "del" or "nabla" operator, represents the gradient of a vector field, which is a vector that points in the direction of the steepest increase of the field at a given point.

What is the significance of the dot product (·) in this equation?

The dot product in this equation represents the combination of two vectors to produce a scalar value, which is used to calculate the work done by a force on a particle. In this context, it is used to calculate the flux of a vector field through a surface.

What is the purpose of the dS term in this equation?

The dS term represents the surface element, which is used to calculate the flux of a vector field through a two-dimensional surface. It is often defined as the cross product of two infinitesimal vectors tangent to the surface.

How is "Calculating doubleintM (∇×F) ·dS" applied in real-world scenarios?

This calculation is commonly used in the study of electromagnetism to determine the electric and magnetic fields produced by various sources. It is also used in fluid dynamics to calculate the flow of fluids through surfaces, such as in pipes or around objects.

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