Let σ be a piecewise smooth oriented surface that is bounded by simple, closed, piecewise smooth curve C with positive orientation. If the component function of vector field F x , y , z have continuous first partial derivatives on some open set containing first partial derivatives on some open set containing σ , and if T is the unit tangent vector to C , then Stokes’ Theorem state that the line integral _ _ _ _ _ and the surface integral _ _ _ _ _ are equal.
Let σ be a piecewise smooth oriented surface that is bounded by simple, closed, piecewise smooth curve C with positive orientation. If the component function of vector field F x , y , z have continuous first partial derivatives on some open set containing first partial derivatives on some open set containing σ , and if T is the unit tangent vector to C , then Stokes’ Theorem state that the line integral _ _ _ _ _ and the surface integral _ _ _ _ _ are equal.
Let
σ
be a piecewise smooth oriented surface that is bounded by simple, closed, piecewise smooth curve C with positive orientation. If the component function of vector field
F
x
,
y
,
z
have continuous first partial derivatives on some open set containing first partial derivatives on some open set containing
σ
,
and if T is the unit tangent vector to C, then Stokes’ Theorem state that the line integral
_
_
_
_
_
and the surface integral
_
_
_
_
_
are equal.
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.
A 20 foot ladder rests on level ground; its head (top) is against a vertical wall. The bottom of the ladder begins by being 12 feet from the wall but begins moving away at the rate of 0.1 feet per second. At what rate is the top of the ladder slipping down the wall? You may use a calculator.
Explain the focus and reasons for establishment of 12.4.1(root test) and 12.4.2(ratio test)
use Integration by Parts to derive 12.6.1
Chapter 15 Solutions
Calculus Early Transcendentals, Binder Ready Version
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