Please answer all questions and parts. Thank you!
Problem 1: Find the flux integral of F=ry-3, ry+1,5 through the parallelogram formed by the vertices A(0,0,1), B(1,1,1), C(6,1,4), D(5,0,4).
Hint: Find the normal vector and the area of a parallelogram using cross products. This is Math 13 material. If done correctly, your integral should become ∮dA for some constant C.
Problem 2: Say whether the following integrals are positive, negative, or zero. Explain why. You don't need to calculate these integrals.
a) The circulation of the vector field around the ellipse x^2+y^2=1 oriented clockwise.
b) The flux coming out of the closed surface z+y+z-1=1 of the vector field <-x,0,->
Problem 3: Show that we can compute the area of a region R in the xy plane bounded by a curve C by finding half of the line integral of the vector field F=-x around the curve C.
Can you give an example of a vector field not equal to F that works instead? Hint: What should ∇ x F equal?