Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface . 45. F = 〈 x , y , z 〉 across the slanted surface of the cone z 2 = x 2 + y 2 , for 0 ≤ z ≤ 1; normal vectors point upward.
Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface . 45. F = 〈 x , y , z 〉 across the slanted surface of the cone z 2 = x 2 + y 2 , for 0 ≤ z ≤ 1; normal vectors point upward.
Surface integrals of vector fieldsFind the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or parametric description of the surface.
45.F = 〈x, y, z〉 across the slanted surface of the cone z2 = x2 + y2, for 0 ≤ z ≤ 1; normal vectors point upward.
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.
Path of steepest descent Consider each of the following surfaces and the point P on the surface.
a. Find the gradient of ƒ.
b. Let C’ be the path of steepest descent on the surface beginning at P, and let C be the projection of C’ on the xy-plane. Find an equation of C in the xy-plane.
c. Find parametric equations for the path C’ on the surface.
ƒ(x, y) = y + x (a plane); P(2, 2, 4)
Fur
Find the area of the surface.
Syr
The helicoid (or spiral ramp) with vector equation r(u, v) = u cos(v) i + u sin(v) j + v k, 0 s u s 1, 0 s vs 5x.
Path of steepest descent Consider each of the following surfaces and the point P on the surface.
a. Find the gradient of ƒ.
b. Let C’ be the path of steepest descent on the surface beginning at P, and let C be the projection of C’ on the xy-plane. Find an equation of C in the xy-plane.
c. Find parametric equations for the path C’ on the surface.
ƒ(x, y) = 4 - x2 - 2y2 (a paraboloid); P(1, 1, 1)
Chapter 17 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
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