Verifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation. 10. F = 〈– y – x , – z, y – x 〉; S is the part of the plane z = 6 – y that lies in the cylinder x 2 + y 2 = 16 and C is the boundary of S.
Verifying Stokes’ Theorem Verify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation. 10. F = 〈– y – x , – z, y – x 〉; S is the part of the plane z = 6 – y that lies in the cylinder x 2 + y 2 = 16 and C is the boundary of S.
Verifying Stokes’ TheoremVerify that the line integral and the surface integral of Stokes’ Theorem are equal for the following vector fields, surfaces S. and closed curves C. Assume that C has counterclockwise orientation and S has a consistent orientation.
10.F = 〈–y –x, –z, y – x〉; S is the part of the plane z = 6 – y that lies in the cylinder x2 + y2 = 16 and C is the boundary of S.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Identify the surface by eliminating the parameters from the vector-valued function
r(u,v) = 3 cosv cosui + 3 cosv sinuj + Śsinvk
a. plane
b. sphere
c. paraboloid
d. cylinder
e. ellipsoid
d
b
a
e
(D
True or False: The vector (10, 2, –4) is normal to the surface x2 + y? – 22 = 22 at the point P = (5, 1,2).
True
O False
Use Stokes' Theorem to evaluate F. dr where F = (x + 7z)i + (10x + y)j + (8y − z) k_and C is the curve
of intersection of the plane x + 3y + z = 18 with the coordinate planes.
(Assume that C is oriented counterclockwise as viewed from above.)
Thomas' Calculus: Early Transcendentals (14th Edition)
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