- #1
BiGyElLoWhAt
Gold Member
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OK, so I know stokes theorem states that I can turn ##\int \int_{S}F\cdot dS## into ##\int \int (-P\frac{\partial g(x,y)}{\partial x} -Q\frac{\partial g(x,y)}{\partial y} + R)dxdy##
but I have a problem that I'm working on that's screaming change of variables (to spherical), but I'm not sure if I can apply stokes' this way...
my region is ##x^2 +y^2 + z^2 = 64##, and I really want to make my region a spherical region (if possible, as I need to understand this for my final) but I'm not sure how to define g.
What's throwing me off is which partials to take. My F is ##<x,-y,z>## and upon making the proper substitutions, ##- \rho <cos(\theta )sin(\phi ), -sin(\theta )sin(\phi ), cos(\ phi)>##
and I suppose g would be ##\rho <cos(\theta )sin(\phi ), sin(\theta )sin(\phi ), cos(\ phi)>## , since that's my region? It's the region in the first quadrant oriented inwards, that's where the negative sign came from, I guess it should probably be in g, but either way it's going make it into the integral.
But now, how do I define my g and my partials? I understand P is my ##\hat{i}## and Q j and R z, but I'm uncertain as to which partials I'm multiplying by which functions.
I'm not sure if I can even do this, I'm just trying to understand this better. any help would be greatly appreciated =]
but I have a problem that I'm working on that's screaming change of variables (to spherical), but I'm not sure if I can apply stokes' this way...
my region is ##x^2 +y^2 + z^2 = 64##, and I really want to make my region a spherical region (if possible, as I need to understand this for my final) but I'm not sure how to define g.
What's throwing me off is which partials to take. My F is ##<x,-y,z>## and upon making the proper substitutions, ##- \rho <cos(\theta )sin(\phi ), -sin(\theta )sin(\phi ), cos(\ phi)>##
and I suppose g would be ##\rho <cos(\theta )sin(\phi ), sin(\theta )sin(\phi ), cos(\ phi)>## , since that's my region? It's the region in the first quadrant oriented inwards, that's where the negative sign came from, I guess it should probably be in g, but either way it's going make it into the integral.
But now, how do I define my g and my partials? I understand P is my ##\hat{i}## and Q j and R z, but I'm uncertain as to which partials I'm multiplying by which functions.
I'm not sure if I can even do this, I'm just trying to understand this better. any help would be greatly appreciated =]