Flux integrals Assume the vector field F = ( f , g ) is source free (zero divergence) with stream function ψ. Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral ∫ C F ⋅ n d s is independent of path; that is, ∫ C F ⋅ n d s = ψ ( B ) − ψ ( A ) .
Flux integrals Assume the vector field F = ( f , g ) is source free (zero divergence) with stream function ψ. Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral ∫ C F ⋅ n d s is independent of path; that is, ∫ C F ⋅ n d s = ψ ( B ) − ψ ( A ) .
Flux integrals Assume the vector field F = (f, g) is source free (zero divergence) with stream function ψ. Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral
∫
C
F
⋅
n
d
s
is independent of path; that is,
∫
C
F
⋅
n
d
s
=
ψ
(
B
)
−
ψ
(
A
)
.
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.
The figure shows the vector field
F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end at (3,2).
(a) Explain why∫c F. dr has the same value for all the three curves.
(b) What is this common value?
Sketch and describe the vector field F (x, y) = (-y,2x)
2. Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such that
[₁
C₁
F. dr = 0 and
[₁
C₂
F. dr = 1
University Calculus: Early Transcendentals (4th Edition)
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