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Homework Statement
[tex]\int_{S}[/tex][tex]\int[/tex] n dS = 0 for any closed surface S.
Homework Equations
The Attempt at a Solution
I can't solve this because I don't have any idea in Vector intregrals.
Hint: The result is a vector, so look at its components. The x component of a vector V is [itex] V \cdot i[/itex]. So look at, for example:Homework Statement
[tex]\int_{S}[/tex][tex]\int[/tex] n dS = 0 for any closed surface S.
Homework Equations
The Attempt at a Solution
I can't solve this because I don't have any idea in Vector intregrals.
Homework Statement
Homework Equations
The Attempt at a Solution
What is i there???Hint: The result is a vector, so look at its components. The x component of a vector V is [itex] V \cdot i[/itex]. So look at, for example:
[tex] i \cdot \int\int_S \hat n\ dS = \int\int_S i\cdot \hat n\ dS[/tex]
So, will you help me to make a proof?i is the unit vector in the x direction, the usual i,j,k notation. n is the unit outward normal.
is there any proof you can show??If the vector field is always tangential to the surface (and therefore perpendicular with the normal of the surface), this relation is trivially true. If the vector field is 0 everywhere, then this relation is also trivially true. What is the question exactly? Find all such vector fields n for which this relation holds?
As LCKurtz already pointed out, [itex]\textbf{n}[/itex] is the outward unit normal to whichever closed surface is integrated over. The problem is to show that this integral is zero for any closed surface.If the vector field is always tangential to the surface (and therefore perpendicular with the normal of the surface), this relation is trivially true. If the vector field is 0 everywhere, then this relation is also trivially true. What is the question exactly? Find all such vector fields n for which this relation holds?
We don't make proofs for you here. LCKurtz has given you a very good hint in his first reply, try using it and show us what you get. You should find that the divergence theorem is very useful to you hereSo, will you help me to make a proof?