Explain the meaning of the integral ∬ S ( ∇ × F ) ⋅ n d S in Stokes’ Theorem.
Explain the meaning of the integral ∬ S ( ∇ × F ) ⋅ n d S in Stokes’ Theorem.
Solution Summary: The author explains the Stokes' Theorem, where S is an oriented surface in R3 with a piecewise-smooth closed boundary C whose orientation is consistent with that of
Explain the meaning of the integral
∬
S
(
∇
×
F
)
⋅
n
d
S
in Stokes’ Theorem.
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.
Find the fluid force (in lb) on a circular observation window of radius 1 foot in a vertical wall of a large water-filled tank at a fish hatchery when the center of the window is 8 feet and d feet (d > 1)
below the water's surface (see figure). Use trigonometric substitution to evaluate the one integral. Water weighs 62.4 pounds per cubic foot. (Recall that in Section 7.7, in a similar problem, you
evaluated one integral by a geometric formula and the other by observing that the integrand was odd.)
(a) 8 feet
(b) d feet below
X
lb
lb
x² + y² = 1
-2
2
X
c) Verify Stokes's Theorem for F = (x²+y²)i-2xyj takes around the rectangle bounded by the lines x=2,
x=-2, y=0 and y=4
Chapter 17 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
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Fundamental Theorem of Calculus 1 | Geometric Idea + Chain Rule Example; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=hAfpl8jLFOs;License: Standard YouTube License, CC-BY