ive done 15 pages of a calc III take home test, and this is the only thing im having trouble with.
so i seem to have taken poor notes when going over the concept of area vectors. i think my handwriting says that for flat surfaces, the area vector is the total area times the normal vector. is this true?
really, my problem is given a vector field f = ai + bj + ck where a,b, and c are constants,
4) let s be the disk of radius 2010 centered on the x-axis, in the plane x = -1, oriented away from the yz-plane. find the condition(s) on the constants a,b,and c such that the flux of f through s i positive or explain why no such condition exists.
5) let s be a triangle with vertices P (1,0,0), Q(0,2,0) and R(0,0,3) oriented away from the origin. same question.
6) let s be the unit sphere centered at (-4,29,-10) oriented outward. same question.
answers (i think)
4) so if i know the area vector, then i can dot it with the vector field to find the flux. once they're crossed, it will be obvious what conditions will allow flux to be positive.
5) the same is true for this one, right? if i know the area vector, i can just dot it?
6) i have no idea since it's not a flat surface. the only thing we have learned is the divergence theorem, but this won't get me anywhere being that the derivatives of a, b, and c must be zero since they are constant.
any help would be greatly appreciated.