2022-02-11
Use the Divergence Theorem to calculate the surface integral F · dS, that is, calculate the flux of F across S.
S is the surface of the solid bounded by the cylinder
alenahelenash
Expert2023-04-23Added 556 answers
To solve this problem, we will use the Divergence Theorem, which relates the flux of a vector field across a closed surface to the divergence of the vector field within the enclosed volume.
First, let's find the divergence of F:
Now, we need to find the surface S, which is the surface of the solid bounded by the cylinder and the planes . This surface consists of three parts: the top and bottom surfaces of the cylinder (which are circles), and the curved surface between them.
We can calculate the flux across each of these surfaces separately and add them up to get the total flux across S. However, since the cylinder and the plane x=0 are perpendicular to the x-axis, the only contribution to the flux across them comes from the x=4 plane.
Therefore, we can write the surface integral as:
where D is the projection of S onto the xy-plane (i.e. the circle with radius 2 centered at the origin), n is the unit normal vector to S, and dA is the area element on D.
To evaluate this integral, we need to find and dA. Since the cylinder is symmetric about the yz-plane, we can choose n to be in the x-direction. Therefore,
To find , we can use polar coordinates: , where r is the radial distance from the origin.
Substituting these expressions into the integral, we get:
Therefore, the flux of F across S is .
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