Let C be the counter-clockwise planar circle with center at the origin and radius r o. VWithout computing them, determine for the following vector fields F whether the line integrals F. dr are positi...
1. Let F-yi(xr +6g) j + 2z k and (a) Which of these two fields (if any) are conservative on R3? Give detailed (b) Find potential functions for the fields that are conservative (c) Calculate the line integrals F dr and G dr where C is the arc of the reasoning r2 4 curve formed by the intersection of the plane z = 4 and the surface--+92 in the first octant, oriented anti-clockwise when view from above.
1. Let F-yi(xr...
3. Consider the vector field F(x, y) + 2y F dr, where C is the circle (r-2)2 +y2 = 1, oriented counterclock (a) Compute wise (Hint: use the FT of line integrals. We could not use it for the circle centered at the origin, but we can use the theorem for this circle. Why?) (b) Let 0 be the angle in polar coordinates for a point (x, y). Check that 0 is a potential function for F
3. Consider the...
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...
10. Stokes' Theorem and Surfac e Integrals of Vector Fields a. Stokes' Theorem: F-dr= b. Let S be th ky-plane. Draw a sketch of curve C in the xy-plane. et be the surface of the paraboloid z 4-x-y and Cis the trace of S in the c Let Fox.y.z) <2z, x, y>, Compute the curl (F) d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) F-dr Use Stokes' Theorem to compute , e.
10. Stokes' Theorem and Surfac...
True or False Determine whet her the statement is true or false, and circle the correct answer. Each question is worth 2 points. (1) If F is a vector field and C is an oriented curve, then F dr must be less than zero. F (2) It is possible that for a certain vector field F and piecewise smooth oriented path C we have/. F. dr-2i-Sj. (3) Suppose d·is the unit square joining the points (0,0), (1,0), (1,1), (0.1) oriented...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...