2. In this problem you will learn how to apply the Stokes theorem. (a) For vector...
4. (18 points) Verify Stokes' Theorem in finding the counterclockwise circulation of the vector field, F - (r-i + (42)j + (r) k around the curve, C, where C is the triangular path determined by the points (6,0,0),(0,-4,0),and (0,0,10) . (i.e. calculate the circulation % F.iF directly, and then by using Stokes' Theorem and doing a surface integral.) Which way was easier? (Hint: You will need to find the equation of the plane that goes through these three points.) 4....
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
(2 pts) Calculate the circulation, rF dr, in two ways, directly and using Stokes' Theorem. The vector field F (8x-8y+62)(i + j) and C is the triangle with vertices (0,0,0), (8, 0, 0), (8,2,0), traversed in that order. Calculating directly, we break C into three paths. For each, give a parameterization r (t) that traverses the path from start to end for 0sts 1 On Ci from (0,0, 0) to (8,0,0), r(t) = <8t,0,0> On C2 from (8, 0, 0)...
a) What does the solenoidal vector field and irrotational vector field mean, what does it mean physically? Show that in a single mathematical expression, a vector field A is solenoidal and irrotational, respectively. b) A solenoidal field vector along the surface integral of a closed surface is equal to 0 to show through the divergence theorem. c) Show by means of the Stokes theorem that the line integral of an irrotational vector field along the closed curve surrounding a surface...
Select statements that are correct. Green's Theorem calculate the circulation in R^2 which convert the line integral into a double integral over the region Din R^2 formed by the simple and closed curve C To compute the work done by a vector field in moving a particle around a simple and closed curve Cin R^2, we apply the Green's Theorem U line integral of a vector field computes the work done to move a particle along a space curve C...
(a) Find the flux of the vector field F=yi-xjtk across the surface σ which is 4. x2 +y2 and below z the portion of z 4 and is oriented by the outward normal. _t7г (b) Use Stokes' Theorem to evaluate the line integral of J F.dr of F--уз ì_x3 j+(x+z)k where C is the clockwise path along the triangle with vertices (0,0,0). (1.0,0)and (1.i.o) aong the thiangle with(i) t) (a) Find the flux of the vector field F=yi-xjtk across the...
17.2 Stokes Theorem: Problem 2 Previous Problem Problem List Next Problem (1 point) Verify Stokes' Theorem for the given vector field and surface, oriented with an upward-pointing normal: F (ell,0,0), the square with vertices (8,0, 4), (8,8,4),(0,8,4), and (0,0,4). ScFids 8(e^(4) -en-4) SIs curl(F). ds 8(e^(4) -e^-4) 17.2 Stokes Theorem: Problem 1 Previous Problem Problem List Next Problem (1 point) Let F =< 2xy, x, y+z > Compute the flux of curl(F) through the surface z = 61 upward-pointing normal....
Calculate the left and right hand side of Stokes Theorem for this problem. Explain why you obtained different values, and why it is not a contradiction 2. [5 Stokes' Theorem: Let S be a piecewise smooth oriented surface having a piecewise smooth boundary curve C. Let F = Mi+Nj+Pk be a vector field whose components have continuous first partial derivatives on an open region containing S. The the circulation of F around C in the direction counterclockwise with respect to...
calc hw- pls help!! (: -/5 POINTS MY NOTES Use Stokes' theorem to evaluate | vxř. ñ ds where F = 9y?z, 6xz, 7x?y2 and S is the paraboloid z = x2 + y2 inside the cylinder x2 + y2 = 1, oriented upward. Submit Answer -/5 POINTS MY NOTES Use Stokes' theorem to compute the circulation F. dr where F = (6xyz, 3y-z, 2yz) and C is the boundary of the portion of the plane 2x + 3y +...
LE 4) (Ungraded) In Cartesian coordinates, the curl of a vector field Air) is defined as Use the definition of electric potential to find the potential difference between the origin and r = x + y + 27, V(r) - V(O) = - Ed. As the line integral is independent of path, choose whatever path you find to be con- vertient Taking V(0) = 0, what is V(r)? Finally, confirm that taking the gradient of the potential recovers our original...