A surface S described as circular paraboloid x2 + y2-z bounded by plane z-3. a) Find a parametrization for S. b) Find plane tangent to S at (1, 1, 2) A surface S described as circular paraboloid...
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y?». Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) e. Use Stokes' Theorem to computec F dr 10. Stokes' Theorem and Surface Integrals...
10. Stokes Theorem and Surface Integrals of Vector Fields a Stokes Theorem:J F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y, Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)ーーーーーーーーーーーーー Compute N(u,v) e. Use Stokes' Theorem to compute Jc F dr 10. Stokes Theorem and Surface...
Evaluate ∫∫ (x2i+ xyj+ zk) ∙ ndA over the closed surface bounded by the paraboloid S: z = 4 − x2−y2, on sides, and the bottom plane z = 0 (a) directly, (b) by the divergence theorem.
Consider the paraboloid z=x2+y2. The plane 2x−2y+z−7=0 cuts the paraboloid, its intersection being a curve. Find "the natural" parametrization of this curve. Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your...
2. Given the paraboloid: z + x2 + y2 = 6. a) Find the symmetric equations of the normal line at the point (1, 2, 1). b) Find the equation of the tangent plane at the point (1, 2, 1). Simplify. c) Find the angle of inclination of the tangent plane to the xy-plane in degrees. Round to the tenths place.
Suppose F(z, y, z) = (z, y, 5z). Let W be the solid bounded by the paraboloid z = x2 + y2 and the plane z = 16. Let S be the closed boundary of W oriented outward. (a) Use the divergence theorem to find the mux of F through S. (b) Find the flux of F out the bottom of S (the truncated paraboloid) and the top of S (the disk).
4. Find the surface area of part of the paraboloid z =x2 + y2 cut of by the plane z = 4.
Verify Stokes, Theorem for the surface S that is the paraboloid given by z = 6-x2-y2 that lies above the plane z 2 (oriented upward) and the vector field F(x, y, z)2yzi+yj+3xk. Verify Stokes, Theorem for the surface S that is the paraboloid given by z = 6-x2-y2 that lies above the plane z 2 (oriented upward) and the vector field F(x, y, z)2yzi+yj+3xk.
Let Surface S be that portion of the paraboloid z = x2 + y2, which lies between the planes z = 4 and z = 25. Find the Area of S.
If someone could please help me out with # 2,3,4. Thank you. 2) the region bounded by the paraboloid z x2 + y2 and the cylinder x2 y2-25 2 2500 1875 2 625 625 3) the region bounded by the cylinderx2+y2 9 and the planes z 0 and x + z 7 A) 637 B) 4417 C) 21π D) 147T 4) the region bounded by the paraboloid z x2+ y2, the cylinderx2 + y2- 81, and the xy-plan 6561 2...