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.
Calculate the vector field flow g⟶ (x,y,z) = (y) î - (x) j + (x + y) k, counterclockwise, along the curve intersecting the surfaces z = x2 + y2 and z = 1. Calculate in two ways:
a) Through direct calculation of the line integral
b) Through Stokes' theorem
Verify Stokes' Theorem for the vector field F(x,y,z) = z^2i - 2xj + y^3k taking the surface S as the upper half of the unit sphere x^2 + y^2 + z^2 = 1.
Thomas' Calculus: Early Transcendentals (14th Edition)
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