Evaluate the outward flux of the vector field F(x,y,z) = (x, -y, z) across the surface S: x2 +y2 =4, 0 <= z <= 1. Use a parametric description of the cylinder surface.

Let S be the surface obtained by revolving the plane curve 2x=sqrt(9−y2) about the y-axis.Obtain a vector equation for S and use it to find the equation of the tangent plane to S at the point (-1,1,-1).

4. Consider the vector function r(z, y) (r, y, r2 +2y"). (a) Re-write this vector function as surface function in the form f(1,y). (b) Describe and draw the shape of the surface function using contour lines and algebraic analysis as needed. Explain the contour shapes in all three orthogonal directions and explain and label all intercepts as needed. (c) Consider the contour of the surface function on the plane z= for this contour in vector form. 0. Write the general equation

Find a vector function r(t) that describes the curve of intersection be- tween the elliptical cylinder 4y? + z² = 4 and the plane x + y = 1. Sketch the curve of intersection by skecthing the two surfaces.

Find a vector parallel to the line of intersection of the planes 5x−y+2z=4 and 4x−y−3z=0v→=  (b) Show that the point (−1,−1,−1) lies on both planes. Then find a vector parametric equation for the line of intersection.r→(t)=