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.
TRUE OR FALSE?
Given an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth boundary curve with positive orientation, and a vector field whose components have continuous partial derivatives on an open region in three-dimensional Euclidean space. Then the line integral of this vector field along the boundary curve equals the flux of the curl of the vector field across the surface
use Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C.
F = (x - y)i + ( y - x)j
C: The square bounded by x = 0, x = 1, y = 0, y = 1
use Green’s Theorem to find the counterclock-wise circulation and outward flux for the field F and curve C.
F = (y2 - x2 )i + (x2 + y2 )j
C: The triangle bounded by y = 0, x = 3, and y =x.
Chapter 15 Solutions
Calculus Early Transcendentals, Binder Ready Version
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.