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
4. Consider the vector function r(z, y) (r, y, r2 +2y").
(a) Re-write this vector function as surface function in the form f(1,y).
(b) Describe and draw the shape of the surface function using contour lines and algebraic analysis
as needed. Explain the contour shapes in all three orthogonal directions and explain and label
all intercepts as needed.
(c) Consider the contour of the surface function on the plane z=
for this contour in vector form.
0. Write the general equation
Thomas' Calculus: Early Transcendentals (14th Edition)
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