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pivoxa15
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Homework Statement
Find the flux of the vector field F(x,y,z)=(z,y,x) across the unit sphere x^2+y^2+z^2=1
The Attempt at a Solution
My Answer: 3(pi)^2/8
Book answer: 4(pi)/3
Dmak said:yes but no need for spherical coordinates since its just the triple integral:
SSS1dV = volume of sphere over the domain d = { (x,y,z): x^2 + y^2 + z^2 = 1 }
( sorry no latex )
Gib Z said:If dexter still posted regularly, I'm sure he would have made a point to say Volume *enclosed by* the sphere =] Welcome to PF Dmak! (And don't mind that comment, really just semantics).
EDIT: Wow Dick has 2^(12) posts =]
Flux through a sphere is a measure of the amount of flow or movement of a vector field through the surface of a sphere. It is a concept used in physics and engineering to understand the flow of various physical quantities, such as electric or magnetic fields, through a three-dimensional space.
The flux through a sphere can be calculated using the formula F = ∫∫S F · dS, where F is the vector field, S is the surface of the sphere, and dS is the differential element of the surface. This integral represents the summation of all the vector field values at each point on the surface, taking into account the direction and magnitude of the field at that point.
The flux through a sphere is affected by the strength and direction of the vector field, the size and shape of the sphere, and the orientation of the sphere with respect to the field. It is also affected by any obstacles or boundaries present in the field that may cause disruptions or changes in the flow.
The concept of flux through a sphere is closely related to Gauss's law, which states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space. This means that the flux through a sphere can be calculated using the electric field and the charge enclosed within the sphere.
The concept of flux through a sphere has many practical applications in fields such as engineering, physics, and meteorology. It is used in the design of electrical circuits, the study of weather patterns, and the analysis of fluid flow in pipes and channels. It is also used in medical imaging techniques, such as MRI, to understand the flow of fluids or particles within the body.