Stokes’ Theorem for evaluating line integrals Evaluate the line integral ∮ C F ⋅ d r by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. 13. F = 〈 x 2 – z 2 , y , 2 xz 〉; C is the boundary of the plane z = 4 – x – y in the first octant.
Stokes’ Theorem for evaluating line integrals Evaluate the line integral ∮ C F ⋅ d r by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation. 13. F = 〈 x 2 – z 2 , y , 2 xz 〉; C is the boundary of the plane z = 4 – x – y in the first octant.
Stokes’ Theorem for evaluating line integralsEvaluate the line integral
∮
C
F
⋅
d
r
by evaluating the surface integral in Stokes’ Theorem with an appropriate choice of S. Assume that C has a counterclockwise orientation.
13. F = 〈x2 – z2, y, 2xz〉; C is the boundary of the plane z = 4 – x – y in the first octant.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Calculate the curl(F) and then apply Stokes' Theorem to compute the flux of curl(F) through the surface of part of the cone
√x² + y2 that lies between the two planes z = 1 and z = 8 with an upward-pointing unit normal, vector using a line
integral.
F = (yz, -xz, z³)
(Use symbolic notation and fractions where needed.)
curl(F) =
flux of curl(F) = [
Use Green's Theorem to evaluate the integral. Assume that the curve C is oriented counterclockwise.
3 In(3 + y) dx -
-dy, where C is the triangle with vertices (0,0), (6, 0), and (0, 12)
ху
3+y
ху
dy =
3 In(3 + y) dx -
3+ y
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