Problem 4: Use the surface integral in Stokes' theorem to evaluate F.dr for the hemisphere S...
step by step please, thank you (2) Use Stokes' Theorem to evaluate the integral F.dr, where F(x, y, z) =< -Y, I, z > and where S is the upper hemispherical surface defined by z = v1- 2 - y2. The boundary of S is the curve C defined by Cos (t) y= sin (t) 0t 27 Z=0
Let F = < x-eyz, xexx, z?exy >. Use Stokes' Theorem to evaluate slice curlĒ ds, where S is the hemisphere x2 + y2 + z2 = 1, 2 > 0, oriented upwards.
Evaluate the surface integral F.ds for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation F(x, y, z) = yi - xj + Szk, S is the hemisphere x2 + y2 + z2 = 4, z 20, oriented downward -8751 x
3. Using Stokes theorem evaluate fa.dr. where A = (x² + y - 4)i + 3ryj + (2x2 + 2?)k and C is the curve bounding the surface S given by (a) the hemisphere IP + y2 + z2 = 16 above the ry plane (b) the paraboloid z = 4 - (z? + y²) above the ry plane.
Question 3 (10 marks) Use Stokes' theorem to evaluate ff(VxG)•dS where G = 2x² yi + 3xy?j + xyzk and S is the hemisphere x2 + y2 + z2 = 4 with z 20.
Verify that the line integral and the surface integral of Stokes Theorem are equal far the following vector field, surface S, and closed curve C. Assume that C has counterlockwise orientation and S has a consistentorientation F = 〈y,-x, 11), s is the upper half of the sphere x2 + y2 +22-1 and C is the circle x2 + y2-1 in the xy-plane Construct the line integral of Stokes' Theorem using the parameterization r(t)= 〈cost, sint, O. for 0 sts2r...
(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...
8. (10 marks) Verify Stokes' Theorem for F = x?i + xyj + zk, where S is the part of the sphere x2 + y2 + z2 = 1 for z > 0.
Use Green's theorem to evaluate line integral F.dr, where F(x, y) = (y2 – x2)i + (x2 + y2)j, and C is a triangle bounded by y = 0, x = 6, and y = x, oriented counterclockwise.
Please explain clearly and show all steps. Thank you. A hemisphere S is defined by x2 +y2+z2=4 on z20. A vector field F =2yi* -xj” +xzkº exists over the surface and around its boundary C. Use Stokes' Theorem to calculate SSs curlF. NºdS.