3. Calculate F. dr for F(x, y) =<x,y> using only one integral where is the path...
3) Given vector field F(x,y,z)=<y, xz,x? >. Find N dr where T is the path around the triangle with vertices (1,0,0),(0,1,0) and (0,0,1) traced counterclockwise (when viewed from above.)
Evaluate the integral. 3 4 [ rwa f(x) dx where f(x) = 15 - x2 if -3 SXO if 0<x<3
(15 pts) 7) Using the Divergence Theorem, calculate the flux integral JSF dĀ where F(x, y, z) =< 2 + x2,r2 + y2y +> and S is the closed cylinder 22 + y2 = 1 with 03:31.
Use spherical coordinates to calculate the triple integral of f(x, y, z) = y over the region x2 + y2 + z2 < 3, x, y, z < 0. (Use symbolic notation and fractions where needed.) S S lw y DV = help (fractions)
1. Let F(x,y,z) =< 32, 5x, – 2y >. Use Stokes's Theorem to evaluate the integral Scurl F.ds, where S is the part of the paraboloid z = x² + y2 that lies below the plane z = 4 with upward- pointing normal vector.
7. Evaluate the circulation integral [/s<= x F) .nds where F(x, y, z) = (x + 3,4+2,2 + y) and S is part of the upper part of the sphere r2 + y2 + 2+ = 25 with 3 <=55(you may use any theorem you find helpful).
4. Calculate the transition moment integral <2|-(Ēūl> for y-polarized light.
Calculate the integral over the given region by changing to polar coordinates: f(x, y) = 16xyl, 2² + y² < 49 Answer:
Question 9 Evaluate the integral f(x) dx where 203 f(x) = for x <1 for x > 1 6 7 4 5 3 O2 11 2
Consider the vector field F (x, y, z) = <y?, z2, x?>. Compute the curl (F). Use Stokes' Theorem to evaluate S. F. dr where C is the triangle (0,0,0), (1,0,0), and (0, 1, 1) oriented counter-clockwise when viewed from above.