Calc 4 review! I need help with any of all of them!
Justify your work, communicate clearly and mathematically. calculation not allowed. Unless decimal approximations are requested, numerical answers must Label your problems clearly and mark the work you want graded/not graded; also, box in your answers. Using techniques from this class, approximate the value of exy - cosx at (.1,2.3) Consider the vector field F(x, y) = (cos(sin x + y) cosx ex, cos(sin x + y) + y) the work done as you traverse the Archimedes spiral (r = Theta) from (x,y) = (0,0) to (x,y)= (2 pi. 0). (Hint: check to see if the vector field is conservative.) Find the absolute maximum and minimum of y ex-z on the ellipsoid9x2+4y2+36z2=36 Compute Consider the region bounded by 2 = 4 - x2 - y2 and 2 = 0. Let S be the boundary of this region. Compute the flux of F = (xy2z - xz, yz + ev, - zev) across S. If F represents a fluid flow, explain what the above computation means. First, sketch and describe the surface 5: r(u. v) - (ucosv, usinv.v,v), where u Second, let f(x. y. z) be the distance function from the point (x. y, z) to the z - axi% Integrate the function f over S. Pick 2 of the following and solve. Compute fc(:2 + yeI)dx+ (ez + yz)dy + (2xz + y2/2 )dz along the curve C: r(t)= 5 cost, sint, cost, sint). > Consider the curve coordinates this is just the wobbly circle r = 3 + cos St). Compute where F - (-y,x)/(x2 + y2). State and prove Clairaut's theorem. State and prove the chain rule. State and prow Green s theorem for simply-connected regions. Bonus if you can prove Green's theorem for non-simply-connected regions
